# Cut and project sets with polytopal window II: linear repetitivity

Henna Koivusalo  School of Mathematics, Fry Building, Woodland Road, Bristol
BS8 1UG, United Kingdom<EMAIL_ADDRESS>https://research-
information.bris.ac.uk/en/persons/henna-l-l-koivusalo and  James J. Walton
School of Mathematical Sciences, Mathematical Sciences Building, University
Park, Nottingham, NG7 2RD, United Kingdom<EMAIL_ADDRESS>https://www.nottingham.ac.uk/mathematics/people/james.walton

###### Abstract.

This paper gives a complete classification of linear repetitivity (LR) for a
natural class of aperiodic Euclidean cut and project schemes with convex
polytopal windows. Our results cover those cut and project schemes for which
the lattice projects densely into the internal space and (possibly after
translation) hits each supporting hyperplane of the polytopal window. Our main
result is that LR is satisfied if and only if the patterns are of low
complexity (property C), and the projected lattice satisfies a Diophantine
condition (property D). Property C can be checked by computation of the ranks
and dimensions of linear spans of the stabiliser subgroups of the supporting
hyperplanes, as investigated in Part I to this article. To define the correct
Diophantine condition D, we establish new results on decomposing polytopal cut
and project schemes to factors, developing concepts initiated in the work of
Forrest, Hunton and Kellendonk. This means that, when C is satisfied, the
window splits into components which induce a compatible splitting of the
lattice. Then property D is the requirement that, for any suitable
decomposition, these factors do not project close to the origin in the
internal space, relative to the norm in the total space. On each factor, this
corresponds to the usual notion from Diophantine Approximation of a system of
linear forms being badly approximable. This extends previous work on cubical
cut and project schemes to a very general class of cut and project schemes. We
demonstrate our main theorem on several examples, and derive some further
consequences of our main theorem, such as the equivalence LR, positivity of
weights and satisfying a subadditive ergodic theorem for this class of
polytopal cut and project sets.

###### Key words and phrases:

Aperiodic order, cut and project, model sets, repetitivity, Diophantine
approximation

###### 2010 Mathematics Subject Classification:

Primary: 52C23; Secondary: 52C45

We gratefully acknowledge the support of the London Mathematical Society,
Scheme 2. The research of JW was partially support by EPSRC grant
EP/R013691/1.

## Introduction

There are currently two main approaches to systematically constructing
aperiodically ordered patterns: by _tiling substitution_ and by the _cut and
project method_. Whilst there is no single mathematical characterisation of
what makes a pattern ‘aperiodically ordered’, there are several important
indicators of it. Some of the most natural attributes considered in this
context are: _diffraction_ , _complexity_ and _repetitivity_.

An aperiodic pattern can satisfy ideal qualitative properties with respect to
these attributes. For example, a completely disordered pattern has no
distinguishable features in its diffraction, but some aperiodic patterns can
have _pure point diffraction_ , making the diffraction spectrum a countable
sum of $\delta$-peaks. The study of spectral properties is in part motivated
by the discovery of _quasicrystals_ [SBGC84], physical materials which have
sharp peaks in their diffraction patterns, indicating order, but with
rotational symmetry prohibiting them having periodically arranged structure.
On measures of complexity, one may ask if the _complexity function_ $p(r)$,
given by the number of distinct patches of radius $r$ up to translation
equivalence, is ‘minimal’. For example, in the setting of polytopal cut and
project sets (see Corollary 2.5) one always has $p(r)\gg r^{d}$, where $d$ is
the dimension of the pattern; in fact, it is conjectured that this holds for
any non-periodic pattern [LP03, Conjecture 1.1] (it is known to hold for all
linearly repetitive patterns [Len04]). Turning to repetitivity, it is of great
interest to know when a pattern is _linearly repetitive_ (LR), so that
$\rho(r)\leq Cr$ for some $C>0$ and sufficiently large $r$. Here, $\rho(r)$ is
the _repetitivity function_ , defined by letting $\rho(r)$ be the smallest
value of $R$ so that every $r$-patch which appears somewhere in the pattern in
fact appears within radius $R$ of every point, up to translation. The work of
Lagarias and Pleasants [LP03] showed that linear repetitivity is a restrictive
and distinctive property of aperiodic patterns which implies other indicators
of high order, such as fast convergence to uniform patch frequencies.
Furthermore, Lagarias and Pleasants showed that linear repetitivity is the
lowest possible aperiodic repetitivity [LP03, Theorem 8.1].

It is easily shown that substitution tilings (with some standard restrictions,
such as finite local complexity and primitivity) are LR, from which it follows
that their complexity is $p(r)\ll r^{d}$, where $d$ is the dimension of the
pattern. However, the problem of identifying substitution tilings with pure
point diffraction is extremely difficult. The _Pisot Conjecture_ states that
$1$-dimensional irreducible, primitive, substitution tilings always have pure
point diffraction. Although some cases have been resolved (for
$\beta$-substitutions [Bar18] and two-letter substitutions [BD02, HS03,
BMST16]), the full conjecture is currently still open, with the situation in
higher dimensions seeming far out of reach.

For cut and project patterns, our understanding is in some sense the reverse.
It is known for the very general class of _regular_ cut and project schemes
(those whose windows have measure zero boundary) that the resulting patterns
have pure point diffraction [Sch00]. However, determining the nature of their
complexity and repetitivity is more difficult. To make progress, it is natural
to begin with the class of cut and project sets with convex polytopal windows,
which includes a good number of examples of interest, such as the
Ammann–Beenker [Bee82] and the Penrose tilings [Pen80, GS87]. We note that for
the Penrose tilings, the window may be taken as the projection of the unit
hypercube in $\mathbb{R}^{5}$, although in this case the scheme has non-dense
projected lattice in the internal space. However, a more suitable model set
scheme with dense projection is available [BG13, BKSZ90], for which the window
may be taken as a union of pentagons, and this produces cut and project
schemes which are mutually locally derivable (MLD, see [BG13]) to those with a
single decagon window. In Part I of the current work [KW20] a general formula
for the polynomial growth rate of the complexity function was given in terms
of the cut and project data (see Theorem 2.2 below), building on the work of
[Jul10]. In this paper we settle the problem of determining exactly which such
polytopal cut and project sets, satisfying one extra mild restriction (weak
homogeneity, see Definition 1.6), satisfy linear repetitivity.

It has been known for some time in the $2$-to-$1$ canonical case that linear
repetitivity is equivalent to the slope of the physical space being _badly
approximable_ or, equivalently, having continued fraction expansion with
bounded entries [MH38, MH40]. Higher dimensional generalisations of this have
been studied from the perspective of substitutions (see, for example [BF11]).
More generally, the case of ‘cubical cut and project sets’, was solved in
[HKW18]. However, the class of patterns given by cubical windows is somewhat
limited. Whilst the techniques of that paper could be pushed in an ad-hoc way
to treat some non-cubical examples (see [HKW18, Section 7]), it remained
unclear how to deal with general polytopal schemes.

In this paper we give a complete characterisation of linear repetitivity for
all polytopal cut and project sets which also satisfy a property which we call
_weakly homogeneous_. The result is sharp, in the sense that it fails without
assuming this property, see Section 9.2.1 (although in the same section we
show that there are linearly repetitive examples which are not weakly
homogeneous). Our characterisation is in terms of two properties C (low
complexity) and D (Diophantine), which shall be explained in more detail
shortly.

######  Theorem A.

For an aperiodic, polytopal, weakly homogeneous cut and project scheme, the
following are equivalent:

  1. (1)

LR;

  2. (2)

C and D.

Property C is the requirement that the complexity function satisfies $p(r)\ll
r^{d}$, where $d$ is the ‘physical’ dimension of the patterns. As already
mentioned, the complexity exponent may be determined from the cut and project
data in the way described in Part I [KW20], from the hyperplane stabiliser
ranks and the dimensions of their linear spans. We derive further important
consequences of this condition in Section 2. In particular, we show that
property C implies a _hyperplane spanning_ condition of the pattern (see
Definition 1.4 and Theorem 2.8).

The Diophantine condition D (see Definition 5.1) is a higher dimensional
analogue of the projected lattice being _badly approximable_. Loosely
speaking, it says that any lattice point $\gamma\in\Gamma$ in the total space
does not get too close to the physical space
$\operatorname{\mathbb{E}_{\vee}}$, relative to the point’s distance
$\|\gamma\|$ from the origin. Equivalently, any point of the projected lattice
$\Gamma_{<}$ in the internal space must be reasonably far from the origin,
relative to the norm of the lattice point it projected from.

It is well-known for general patterns (not just cut and project sets) that
property C is necessary for LR. So our main theorem shows that property D is
the crux to establishing linear repetitivity for this class of patterns: if C
is assumed (as well as weak homogeneity) then it is surprisingly _only the
projected lattice_ $\Gamma_{<}$, and none of the remaining data, which
determines whether or not the scheme is LR. So to find linearly repetitive
schemes, one needs to look for Diophantine projections of lattices. The next
step is to identify codimension $1$ subspaces in internal space which
intersect $\Gamma_{<}$ with high rank. If there are enough such subspaces to
constitute the boundary of a polytope, with the subspaces intersecting the
projected lattice with sufficiently large ranks, then translates of these
subspaces can be used as supporting hyperplanes of windows of LR cut and
project schemes.

###  0.1. Further details on properties C and D

To precisely quantify what is meant by a projected lattice point being ‘close’
to the origin, one needs to use the correct Diophantine exponent. When the cut
and project set has what is known as ‘constant hyperplane stabiliser ranks’,
property D says the following:

_There is some $c>0$ so that for all $\gamma\in\Gamma\setminus\\{0\\}$ we
have_

$\|\gamma_{<}\|\geq c\cdot\|\gamma\|^{-\delta},\text{ where
}\delta=\frac{d}{n}.$

In the above, $d$ is the ‘dimension’ $\dim(\operatorname{\mathbb{E}_{\vee}})$
and $n$ is the ‘codimension’ $\dim(\operatorname{\mathbb{E}_{<}})$. The
notation of $\Gamma$ for the lattice in the total space
$\operatorname{\mathbb{E}}$, $\operatorname{\mathbb{E}_{\vee}}$ for the
physical space, $\operatorname{\mathbb{E}_{<}}$ for the internal space and
$\gamma_{<}$ for the projection of $\gamma$ to $\operatorname{\mathbb{E}_{<}}$
follows the notation of Part I [KW20]. The Ammann–Beenker example below is a
cut and project set for which the condition of constant hyperplane stabiliser
ranks holds.

######  Example 0.1.

Figure 1. Windows for the Ammann–Beenker (left) and Golden Octagonal (right)
cut and project schemes. Dots and edges are the projections to
$\operatorname{\mathbb{E}_{<}}$ of the vertices and edges, respectively, of
the unit hypercube in $\mathbb{R}^{4}$. The origin is indicated with a larger
dot, along with the projections of the $4$ standard basis vectors.

To illustrate briefly how the above theorem may be applied in practice, we
consider the Ammann–Beenker cut and project scheme (which is known to satisfy
LR because it can also be generated through substitution). The octagonal
window in the internal space is depicted in Figure 1, together with the origin
and projections of the $4$ standard basis vectors of $\mathbb{R}^{4}$ in the
total space, which generate the lattice $\Gamma$. There are $4$ distinct
supporting hyperplanes up to translation, with each parallel subspace
intersecting the projected lattice $\Gamma_{<}$ with rank $2$ (for example,
the vertical subspace contains the projected lattice element pointing South,
as well as the element given by the sum of those pointing NE and NW). Using
the main result of [KW20] (see Theorem 2.2 below), or indeed the earlier
result of Julien from [Jul10], it follows that these patterns satisfy C, that
is, $p(r)\ll r^{2}$. This also shows that the window has constant stabiliser
rank, which also follows (Corollary 3.9) from C and the fact that the window
is not indecomposable (Definition 3.2), which in codimension 2 simply means it
is not a parallelogram (Example 3.4).

Since the scheme has constant hyperplane rank (see Section 3), to establish D
one just needs to show that $\Gamma_{<}$ is Diophantine, as defined above.
Given $\gamma_{<}\in\Gamma_{<}$, we have

$\gamma_{<}=\begin{pmatrix}(n_{4}-n_{2})\sqrt{2}+n_{1}\\\
(n_{2}+n_{4})\sqrt{2}-n_{3}\end{pmatrix},$

where the $n_{i}\in\mathbb{Z}$ are the coordinates of $\gamma\in\Gamma$ which
projects to $\gamma_{<}$. Since $\sqrt{2}$ is badly approximable (it is a
quadratic irrational, so has periodic and hence bounded continued fraction
expansion), the above vector has norm at least $c/\|\gamma\|$, for $\gamma\neq
0$. This establishes that $\Gamma_{<}$ is Diophantine, so D holds and the
Ammann–Beenker tilings are linearly repetitive by Theorem A. More details of
this computation, and similar ones such as for the _golden octagonal tilings_
[BF17] (with window on the right of Figure 1) are given in Section 9.2.2.

###  0.2. Organisation of paper

In Section 1 we set out some important notation and recall the basic setup for
polytopal cut and project schemes. We also introduce here some new conditions
for a cut and project schemes important for our main results: being
_hyperplane spanning_ (Definition 1.4), _homogeneous_ (Definition 1.5) and
_weakly homogeneous_ (Definition 1.6). The weakly homogeneous condition is not
necessary for one direction of our main theorem, in showing that LR implies C
and D, but it is needed in the other direction. The hyperplane spanning
condition will be used extensively, especially when we come to analyse the
shapes of cut regions in Section 7. However, we are able to remove this
assumption from our main theorem by establishing that it always holds for cut
and project schemes satisfying C, in Theorem 2.8. This is done in Section 2,
which contains results on low complexity schemes.

Unlike in the Ammann–Beenker example above, cut and project sets can behave
like sums of schemes, with factors of varying dimension and codimension.
Because of such examples, finding the correct Diophantine property sometimes
requires decomposing the scheme into factors. After decomposing enough, the
window always splits to subsystems with constant stabiliser ranks, allowing us
to derive the correct Diophantine exponents for them. We develop a theory for
cut and project scheme decompositions in Section 3. This extends work of
Forrest, Hunton and Kellendonk [FHK02], who first identified one definition of
such a decomposition. We provide a similar, equivalent definition, which does
not require a choice of inner product (as well as defining the notion of a
decomposition of more components). We also demonstrate that there is an
equivalent geometric description: a decomposition corresponds to a direct sum
decomposition of the window. We also give a simpler proof (using a different
combinatorial graph associated to the flags) that when C is satisfied,
indecomposable windows have constant stabiliser rank (Corollary 3.9). The
geometric factorisation of the window is difficult to apply in isolation,
since the dynamics under consideration rely on the interaction between the
acting lattice and the window. Fortunately, when property C is assumed and the
window has a decomposition, the lattice must also split compatibly as a direct
sum, up to finite index (Theorem 3.18). For a summary of these results, see
Corollary 3.24.

In Section 4 we show how properties of the cut and project scheme relate to
those of the subsystems of a decomposition. In particular, we show how the cut
regions and acceptance domains are related to products of those in the
subsystems (Lemmas 4.4 and 4.8).

In Section 5 we introduce the Diophantine condition D. We develop a base-free
approach for a Diophantine condition for a general, densely embedded lattice.
It is analogous to the badly approximable condition for a real number or, more
generally, a set of vectors in some Euclidean space (see Example 5.2). Before
applying the condition to cut and project schemes, we establish its basic
properties and implications, such as that of ‘transference’, that is, that
Diophantine lattices fill space densely (Theorem 5.8).

In Section 6 we prove that LR implies C and D. This is the easier direction of
the main theorem, in part because only consideration of patch frequencies (but
not uniform spacing of patches) is required. We recall the _positivity of
weights_ (PW) condition (see [BBL13]) and show that, in fact, PW is sufficient
for C and D. This is done by constructing patches of low frequency if either
of these conditions fails; in the case that C fails, the growth rate of
patches is too high for all patches to have low frequencies, and if D fails
then we use short projections of lattice vectors to construct small acceptance
domains.

In Section 7 we establish a close connection between the set of possible
locations of vertices of cut regions with the projected lattice, for (weakly)
homogeneous schemes. This has the implication that if the projected lattice
vectors cannot be too small, then neither are displacements between vertices
of cut regions. Hence, weakly homogeneous cut and project schemes satisfying C
and D have ‘large’ acceptance domains. In Section 8, we combine this
observation with transference from Section 5, to ensure that sufficiently
large orbits are dense enough in internal space to hit all required acceptance
domains, establishing that C and D together imply LR for weakly homogeneous
schemes.

Finally, in Section 9, we consider some implications of our main theorem. We
also demonstrate it on a range of examples.

###  0.3. Summary of notation

  * •

$\mathcal{S}=(\operatorname{\mathbb{E}},\operatorname{\mathbb{E}_{\vee}},\operatorname{\mathbb{E}_{<}},\Gamma,W)$
: Euclidean cut and project scheme $\mathcal{S}$, with total space
$\operatorname{\mathbb{E}}$ of dimension $k$, physical space
$\operatorname{\mathbb{E}_{\vee}}$ of dimension $d$, complementary internal
space $\operatorname{\mathbb{E}_{<}}$, lattice
$\Gamma\leqslant\operatorname{\mathbb{E}}$ and window
$W\subset\operatorname{\mathbb{E}_{<}}$ (always here a convex polytope).

  * •

$\pi_{\vee}\colon\operatorname{\mathbb{E}}\to\operatorname{\mathbb{E}_{\vee}}$
: the projection from the total space to the physical space with respect to
$\operatorname{\mathbb{E}_{<}}$. For $x\in\operatorname{\mathbb{E}}$ we let
$x_{\vee}\coloneqq\pi_{\vee}(x)$. Analogous notation for
$\operatorname{\mathbb{E}_{<}}$ in place of
$\operatorname{\mathbb{E}_{\vee}}$.

  * •

$y^{\wedge}$ : for $y=\pi_{\vee}(\gamma)$, with $\gamma\in\Gamma$, we let
$y^{\wedge}\coloneqq\gamma$.

  * •

$y^{*}$ : the star map applied to $y\in\Gamma_{\vee}$ is
$y^{*}\coloneqq(y^{\wedge})_{<}$.

  * •

$\operatorname{\Lambda}$ : the cut and project set associated to the scheme,
$\operatorname{\Lambda}\coloneqq\\{y\in\Gamma_{\vee}\mid y^{*}\in W\\}$.

  * •

$A_{P}$ : the acceptance domain in $W$ associated to (pointed) patch $P$ of
$\operatorname{\Lambda}$.

  * •

$\Gamma(r)$: those lattice points that are within radius $r$ from the origin.

  * •

$\mathscr{A}(r)$ : the set of acceptance domains of patches of radius $r$.

  * •

$\mathscr{H}^{+}$ : the set of half spaces defining a polytopal window $W$,
with associated collection of affine hyperplanes $\mathscr{H}$, called the
_supporting hyperplanes_ of the window.

  * •

$V(H)$ : the linear subspace given by translating an affine hyperplane $H$
over the origin.

  * •

$\mathscr{H}_{0}$ : the supporting hyperplanes translated over the origin,
that is, $\mathscr{H}_{0}=V(\mathscr{H})$.

  * •

$\Gamma^{H}$ : the stabiliser of $H$ is the subgroup
$\Gamma^{H}\coloneqq\\{\gamma\in\Gamma\mid H=H+\gamma_{<}\\}$ of $\Gamma$.
More generally, given $I\subseteq\mathscr{H}$, we have
$\Gamma^{I}\coloneqq\\{\gamma\in\Gamma\mid H=H+\gamma_{<}\text{ for all }H\in
I\\}=\bigcap_{H\in I}\Gamma^{I}$.

  * •

$(X_{i},\Gamma_{i},W_{i})$: the scheme splits into $m$ subsystems
$(X_{i},\Gamma_{i},W_{i})$, where
$\operatorname{\mathbb{E}_{<}}=X_{1}+\cdots+X_{m}$, $W=W_{1}+\cdots+W_{m}$,
$\Gamma_{1}+\cdots+\Gamma_{m}$ is finite index in $\Gamma$ and each subsystem
has constant hyperplane stabiliser rank. See Section 3 and in particular
Corollary 3.24.

  * •

$\mathcal{V}(G)$ : the set of ‘vertex’ intersections of $G_{<}$ translates of
hyperplanes, where $G\subseteq\Gamma$.

  * •

$\mathcal{V}(G,f)$ : those vertices coming from a flag $f$ of hyperplanes.
Similar notation $\mathcal{V}_{i}(G)$ and $\mathcal{V}_{i}(G,f)$ used for
analogous sets in the subsystems coming from a decomposition. See Definition
7.1.

  * •

$f\ll g$ : for two functions $f,g\colon\mathbb{R}_{>0}\to\mathbb{R}_{>0}$, we
let $f\ll g$ mean that there exists some constant $C>0$ for which $f(x)\leq
Cg(x)$ for sufficiently large $x$. That is, $f\in O(g)$.

  * •

$f\asymp g$ : for two functions as above, $f\ll g$ and $g\ll f$.

##  1\. Basic notions for polytopal cut and project set schemes

A polytopal $k$-to-$d$ cut and project scheme is a tuple

$\mathcal{S}=(\operatorname{\mathbb{E}},\operatorname{\mathbb{E}_{\vee}},\operatorname{\mathbb{E}_{<}},\Gamma,W),$

of a total space, physical space, internal space, lattice and window,
respectively. The total space $\operatorname{\mathbb{E}}$ is a $k$-dimensional
$\mathbb{R}$-vector space, with $\operatorname{\mathbb{E}_{\vee}}$ and
$\operatorname{\mathbb{E}_{<}}$ subspaces of $\operatorname{\mathbb{E}}$ of
dimensions $d$ and $n=k-d$, respectively. We assume that $d$, $n>0$, and that
$\operatorname{\mathbb{E}_{\vee}}$, $\operatorname{\mathbb{E}_{<}}$ are
complementary, giving a direct sum decomposition
$\operatorname{\mathbb{E}}=\operatorname{\mathbb{E}_{\vee}}+\operatorname{\mathbb{E}_{<}}$
of the total space. The lattice $\Gamma\leqslant\operatorname{\mathbb{E}}$ is
a free Abelian group of rank $k$ which is co-compact in
$\operatorname{\mathbb{E}}$. The window
$W\subset\operatorname{\mathbb{E}_{<}}$ is chosen to be a compact, convex
polytope, so it is given as the intersection of a uniquely defined irredundant
set of half-spaces $\mathscr{H}^{+}$. The set of associated codimension $1$
supporting hyperplanes (the boundaries of the half spaces of
$\mathscr{H}^{+}$) is denoted by $\mathscr{H}$.

By a hyperplane, we mean a codimension $1$ affine hyperplane (and say
‘codimension $1$ subspace’ if we wish to emphasise that the hyperplane
contains the origin). Given a hyperplane $H$, we let $V(H)$ denote the
codimension $1$ subspace given by shifting $H$ over the origin, that is
$V(H)=H-H=H-h$ for any $h\in H$. By a slight abuse of notation, we sometimes
use $V$ (and variations) as also the name of some codimension $1$ subspace; we
use $H$ (and variations) for affine hyperplanes. We introduce the notation
$\mathscr{H}_{0}=V(\mathscr{H})$ for the set of subspaces parallel to the
supporting hyperplanes.

We recall the notation
$\pi_{\vee}\colon\operatorname{\mathbb{E}}\to\operatorname{\mathbb{E}_{\vee}}$
for the projection to the physical space, with respect to the direct sum
decomposition
$\operatorname{\mathbb{E}}=\operatorname{\mathbb{E}_{\vee}}+\operatorname{\mathbb{E}_{<}}$.
For a point $X\in\operatorname{\mathbb{E}}$ (or a subset
$X\subseteq\operatorname{\mathbb{E}}$), we let
$X_{\vee}\coloneqq\pi_{\vee}(X)$. Analogous notation is used for projections
to the internal space. This notation is remembered by visualising a $2$-to-$1$
scheme in the first quadrant, where we project ‘down’ $\vee$ to the physical
space, and ‘left’ $<$ to the internal space.

We assume that $\pi_{\vee}$ is injective on $\Gamma$ and that $\Gamma_{<}$ is
dense in $\operatorname{\mathbb{E}_{<}}$. In addition, we assume throughout
that $\mathcal{S}$ produces aperiodic patterns. Equivalently, $\pi_{<}$ is
injective on $\Gamma$, that is, $\operatorname{rk}(\Gamma_{<})=k$. Given
$x\in\Gamma_{\vee}$, we denote the lift of $x$ back to the lattice by
$x^{\wedge}$, which is the unique point with $(x^{\wedge})_{\vee}=x$. We have
the star map $x\mapsto x^{*}$, defined by $x^{*}\coloneqq(x^{\wedge})_{<}$,
mapping $\Gamma_{\vee}$ into $\operatorname{\mathbb{E}_{<}}$.

The cut and project set produced by $\mathcal{S}$ is denoted

$\operatorname{\Lambda}\coloneqq\\{x\in\Gamma_{\vee}\mid x^{*}\in W\\}.$

The above is appropriate only when the scheme $\mathcal{S}$ is non-singular,
meaning that $x^{*}\notin\partial W$ for all $x\in\Gamma_{\vee}$
(equivalently, $\Gamma\cap(\operatorname{\mathbb{E}_{\vee}}+\partial
W)=\emptyset$ or $(\Gamma_{<})\cap\partial W=\emptyset$). We may always
translate the window to make it non-singular, and any non-singular scheme will
generate cut and project sets with the same sets of finite patches, by density
of $\Gamma_{<}$. In particular, we need only consider the above choice of
$\Lambda$ in determining whether or not the family of cut and project sets
produced by $\mathcal{S}$ are linearly repetitive (this is the family of
patterns given similarly using non-singular translates of the lattice, and
singular patterns given as limits of non-singular ones).

It will be important to keep track of the subgroups of $\Gamma$ which
stabilise various sets of supporting hyperplanes. Given
$I\subseteq\mathscr{H}$, we let

$\Gamma^{I}\coloneqq\\{\gamma\in\Gamma\mid H+\gamma_{<}=H\text{ for all }H\in
I\\}.$

It is easy to see that each $\Gamma^{I}$ is a subgroup of $\Gamma$ and we
denote the rank $\operatorname{rk}(\Gamma^{I})$ by $\operatorname{rk}(I)$. For
a singleton, we simplify notation to $\Gamma^{H}\coloneqq\Gamma^{\\{H\\}}$ and
$\operatorname{rk}(H)\coloneqq\operatorname{rk}(\\{H\\})$. Clearly a
translation stabilises a hyperplane $H$ if and only if it stabilises any other
translate $H+x$, for example $V(H)$. So we may write

$\Gamma^{I}=\\{\gamma\in\Gamma\mid V+\gamma_{<}=V\text{ for all }V\in
V(I)\\}=\\{\gamma\in\Gamma\mid\gamma_{<}\in V\text{ for all }V\in V(I)\\}.$

Hence, after projecting $\Gamma^{I}$ to the internal space (which determines
$\Gamma^{I}$, by injectivity of $\pi_{<}$), we may alternatively describe it
as the subset of $\Gamma_{<}$ contained in the subspace intersection of all
$V\in V(I)$:

######  Lemma 1.1.

For $I\subseteq\mathscr{H}$, $\Gamma^{I}$ is a subgroup of $\Gamma$, whose
projection to $\operatorname{\mathbb{E}_{<}}$ is
$\Gamma^{I}_{<}=\Gamma_{<}\cap X_{I}$, for the subspace $X_{I}=\bigcap_{H\in
I}V(H)$. ∎

###  1.1. Patches, acceptance domains and cut regions

The $r$-patch centred at $y\in\Lambda$ is the subset $P(y,r)\coloneqq
B_{r}(y)\cap\Lambda$, marked with the centre $y$. We typically consider
$r$-patches up to translation equivalence. The complexity function
$p\colon\mathbb{R}_{>0}\to\mathbb{N}$ is defined by letting $p(r)$ be the
number of $r$-patches, up to translation equivalence. Again, this does not
depend on the choice $\operatorname{\Lambda}$ of cut and project set produced
by the scheme $\mathcal{S}$, as long as $\mathcal{S}$ is non-singular.

One of the main results of [KW20] was that for any polytopal scheme, we have
$p(r)\asymp r^{\alpha}$, for some $\alpha\in\mathbb{N}$ which can be
determined by the ranks of the subgroups $\Gamma^{H}$, for $H\in\mathscr{H}$
and the dimensions of their linear spans (in particular, we recall this as
Theorem 2.2 later). By [KW20, Corollary 6.2], $\alpha\leq nd$ (where
$n=k-d=\dim\operatorname{\mathbb{E}_{<}}$ denotes the ‘codimension’). We
define the following property of minimal complexity:

######  Definition 1.2.

We say that the scheme $\mathcal{S}$ satisfies property C, of having minimal
complexity, if $p(r)\ll r^{d}$, that is, there exists some $C>0$ for which
$p(r)\leq Cr^{d}$ for all $r\geq 1$.

We will see in Corollary 2.5 that $\alpha\geq d$ for an aperiodic scheme
$\mathcal{S}$, so we always have $p(r)\gg r^{d}$ and hence C implies that
$p(r)\asymp r^{d}$. We shall explore further useful consequences of this
condition in Section 2.

Given some $r>0$, we consider the lattice points $\gamma\in\Gamma$ for which
$\|\gamma_{\vee}\|$, $\|\gamma_{<}\|\leq r$. Without loss of generality, we
can choose a metric on $\operatorname{\mathbb{E}}$ so that, equivalently,
$\|\gamma\|\leq r$ (and even if we do not, we have such an identification up
to linear rescalings of $r$, which is all that is needed for our results
here). Consider the elements of $\Gamma(r)\coloneqq B_{r}(0)\cap\Gamma$, and
further, denote $\Gamma_{<}(r)\coloneqq\Gamma(r)_{<}$.

Recall that to each $r$-patch $P=P(y,r)$, there is a subset $A_{P}\subseteq
W$, called here an $r$-acceptance domain, such that $P(y,r)=P(z,r)+(y-z)$ if
and only if $z^{*}\in A_{P}$. In particular, the $r$-acceptance domain
$A\subset W$ containing the point $z^{*}\in W$ (for $z^{*}\notin\partial
W+\Gamma_{<}$) is the intersection of those $\Gamma_{<}(r)$ translates of the
interior $\mathring{W}$ and complement $W^{c}$ that contain $z^{*}$ [KW20,
Corollary 3.2]. The set of $r$-acceptance domains is denoted $\mathscr{A}(r)$.

We also have the cut regions, defined as follows. Consider the set
$\mathscr{H}+\Gamma_{<}(r)$ of translates of the supporting hyperplanes.
Removing them from the window splits it into convex connected components,
called $r$-cut regions. The set of $r$-cut regions is denoted
$\mathscr{C}(r)$. Even without further assumptions on the polytopal scheme, we
have the following:

######  Lemma 1.3 (Corollary 4.3 of [KW20]).

For sufficiently large $r$, the $r$-cut regions refine the $r$-acceptance
domains. That is, for all $A\in\mathscr{A}(r)$, with $r$ sufficiently large,
there exists some $C\in\mathscr{A}(r)$ with $C\subseteq A$; equivalently, if
the $r$-patches at $x$, $y\in\operatorname{\Lambda}$ disagree then $x^{*}$ and
$y^{*}$ are separated by a hyperplane $H+\gamma_{<}$, with $H\in\mathscr{H}$
and $\gamma\in\Gamma\cap B_{r}(0)$.

We will use the above in showing that conditions C and D together imply LR:
since cut regions refine acceptance domains, it will be sufficient to show
that all cut regions are visited frequently to derive the same for the
acceptance domains, and hence that all patches appear frequently enough in our
patterns. To obtain a kind of converse to the above lemma (up to linear
rescalings of $r$), so as to refine cut regions by acceptance domains, one
needs to assume further conditions, a very general one being the
_quasicanonical_ condition introduced in [KW20]. For our purposes, however,
this will not be necessary, and we work directly with the acceptance domains
for the converse direction of our main theorem.

###  1.2. Linear repetitivity

For $r>0$, we define $\rho(r)$ to be the smallest value of $R$ for which, for
any $z\in\Lambda$ and any $r$-patch $P$ in $\Lambda$, we have that $P=P(y,r)$,
up to translation, for some $y\in B_{R}(z)$. That is, all $r$-patches which
appear somewhere in $\Lambda$, in fact appear within radius $\rho(r)$ of any
point of $\Lambda$, up to translation.

We say that $\Lambda$ is linearly repetitive if $\rho(r)\ll r$, that is, there
exists some $C>0$ such that $\rho(r)\leq Cr$ for sufficiently large $r$. If
$\Lambda$ is linearly repetitive, then so is any other cut and project set in
the family produced by the scheme $\mathcal{S}$, and we say that $\mathcal{S}$
has property LR.

###  1.3. New conditions

We introduce here some new conditions on the scheme $\mathcal{S}$ which will
be useful in establishing our main theorem. The following will be vital in
Section 7 when we come to compare projected lattice points with vertices of
acceptance domains, as well as calculating ranks of hyperplane stabiliser
groups for low complexity schemes. It will be shown in Theorem 2.8 that this
condition always holds if $\mathcal{S}$ satisfies C.

######  Definition 1.4.

Call $\mathcal{S}$ hyperplane spanning if, for each $H\in\mathscr{H}$, the
linear span of $\Gamma_{<}^{H}$ is of dimension $n-1$, that is,
$\langle\Gamma^{H}_{<}\rangle_{\mathbb{R}}=V(H)$.

The next property asks that all supporting hyperplanes are well-positioned
with respect to the lattice, in the sense that they may be translated by
lattice elements over a common origin. The term _homogeneous_ is motivated by
the codimension 1 case (see Subsection 9.2.1), where the condition allows us
to analyse sizes of acceptance domains via homogeneous (rather than
inhomogeneous) Diophantine properties of the lattice.

######  Definition 1.5.

Call $\mathcal{S}$ homogeneous if there exists some ‘origin’
$o\in\operatorname{\mathbb{E}_{<}}$ for which, for each $H\in\mathscr{H}$,
there exists some $\gamma_{H}\in\Gamma$ so that $o\in H+(\gamma_{H})_{<}$.

Obviously the above property is not affected by translating the window. So an
equivalent condition is that, for some translate of the window,
$H\cap\Gamma_{<}\neq\emptyset$ for each $H\in\mathscr{H}$, as seen by
translating $W$ by $-o$, which re-centres at the origin. For convenience, we
shall usually take $o$ to be the origin of $\operatorname{\mathbb{E}_{<}}$
when assuming homogeneity, which simplifies proofs. Whilst this technically
makes the scheme singular with respect to the untranslated lattice, we
ultimately only require homogeneity to derive affine conditions of the window,
which do not depend on a choice of ‘origin’, such as sizes of acceptance
domains.

For the statement of our main theorem, we only need the following weaker
version of homogeneity. As above, the property is invariant up to translations
of the window, so we may always take $o$ as the origin after an appropriate
translation of $W$.

######  Definition 1.6.

Call $\mathcal{S}$ weakly homogeneous if there exists some ‘origin’
$o\in\operatorname{\mathbb{E}_{<}}$ for which, for each $H\in\mathscr{H}$,
there exists some $\gamma_{H}\in\Gamma$ and $n_{H}\in\mathbb{N}$ so that $o\in
H+(1/n_{H})\cdot(\gamma_{H})_{<}$.

######  Lemma 1.7.

The scheme $\mathcal{S}$ is weakly homogeneous if and only if there exists
some $N\in\mathbb{N}$ so that the scheme $\mathcal{S}^{\prime}$, given by
replacing $\Gamma$ with $(1/N)\cdot\Gamma$, is homogeneous.

###### Proof.

Suppose that $\mathcal{S}^{\prime}$ is homogeneous, and take
$H\in\mathscr{H}$. Then there exists $(1/N)\cdot\gamma_{H}\in(1/N)\cdot\Gamma$
with $o\in H+((1/N)\cdot\gamma_{H})_{<}=H+(1/N)\cdot(\gamma_{H})_{<}$. It
follows that $\mathcal{S}^{\prime}$ is weakly homogeneous, taking each
$n_{H}=N$.

Conversely, suppose that $\mathcal{S}$ is weakly homogeneous, and define
$N\coloneqq\prod_{H\in\mathscr{H}}n_{H}$. For each $H\in\mathscr{H}$, there
exists some $\gamma_{H}\in\Gamma$ so that $o\in
H+(1/n_{H})\cdot(\gamma_{H})_{<}=H+((1/N)\cdot(N/n_{H})\cdot\gamma_{H})_{<}$.
Since $(1/N)\cdot(N/n_{H})\cdot\gamma_{H}\in(1/N)\cdot\Gamma$ for each
$H\in\mathscr{H}$, we see that $\mathcal{S}^{\prime}$ is homogeneous. ∎

##  2\. Minimal complexity schemes

In this section we determine consequences of the condition C of minimal
complexity. In [KW20] we determined the exponent of the complexity function
for general polytopal cut and project schemes. We recall the following
definition, followed by the result [KW20, Theorem 7.1] (where we simplify the
original notation from $\alpha^{\prime}$ and $\alpha_{f}^{\prime}$ to $\alpha$
to $\alpha_{f}$):

######  Definition 2.1.

A set $f=\\{H_{1},\ldots,H_{n}\\}$ of affine, codimension $1$ hyperplanes is
called a flag if their intersection is a singleton.

######  Theorem 2.2.

Consider the collection $\mathscr{F}$ of flags of the supporting hyperplanes
$\mathscr{H}$ of a given polytopal cut and project scheme. Then $p(r)\asymp
r^{\alpha}$ where

$\alpha\coloneqq\max_{f\in\mathscr{F}}\alpha_{f},\ \ \mathrm{for}\ \
\alpha_{f}\coloneqq\sum_{H\in f}(d-\operatorname{rk}(H)+\beta_{H}),\
\text{and}\ \
\beta_{H}\coloneqq\dim\langle\Gamma_{<}^{H}\rangle_{\mathbb{R}}.$

Since $\Gamma_{<}^{H}\leqslant V(H)$, and the latter is a codimension $1$
subspace of $\operatorname{\mathbb{E}_{<}}$, we have that $\beta_{H}\leq n-1$,
with equality for each $H\in\mathscr{H}$ if and only if $\mathcal{S}$ is
hyperplane spanning. By the following lemma, $\alpha\geq d$ if the scheme is
hyperplane spanning (recall that we always assume that $\mathcal{S}$ is
aperiodic).

The main theorem of this section is that if $\mathcal{S}$ satisfies property C
(that is, $\alpha\leq d$) then the scheme is hyperplane spanning, and hence
$\alpha=d$. This is to be expected, since a scheme is of low complexity when
the stabilisers $\Gamma^{H}$ are ‘large’, and one would hope therefore at
least spanning. The difficulty is that a priori it is feasible that each
$\Gamma^{H}$ could have large rank but possibly concentrated in lower
dimensional subspaces. Indeed, schemes can be chosen so that this happens for
several hyperplanes $H$. However, we will show (loosely speaking) that this
cannot happen for enough hyperplanes as to constitute a flag. To establish
this, we will need a few technical results.

######  Lemma 2.3.

Suppose that $\mathcal{S}$ is hyperplane spanning. Then $\alpha_{f}\geq d$ for
each $f\in\mathscr{F}$. In particular $\alpha\geq d$.

###### Proof.

Let $f\subseteq\mathscr{H}_{0}$ be any flag and consider the group

$T\coloneqq\bigoplus_{H\in f}(\Gamma/\Gamma^{H}).$

There is a natural homomorphism $\Gamma\to T$, given by the diagonal embedding
followed by taking the quotients by $\Gamma^{H}$ in each component. This map
is an injection. Indeed, suppose that $\gamma\in\Gamma$ maps to $0$ in each
component, so that $\gamma\in\Gamma^{H}$ for all $H\in f$; equivalently
$\gamma_{<}\in V$ for all $V\in V(f)$. Since $f$ is a flag, the only point in
every $V\in V(f)$ is the origin, so $\gamma_{<}=0$. By aperiodicity, we have
that $\gamma=0$.

It follows that
$k=\operatorname{rk}(\Gamma)\leq\operatorname{rk}(T)=\sum_{H\in
f}(k-\operatorname{rk}(H))$. By hyperplane spanning, each $\beta_{H}=n-1$.
Hence

$\alpha_{f}=\sum_{H\in f}(k-\operatorname{rk}(H)-1)=\left(\sum_{H\in
f}(k-\operatorname{rk}(H))\right)-n=\operatorname{rk}(T)-n\geq k-n=d,$

as required. Since each $\alpha_{f}\geq d$, we have that
$\alpha=\max_{f\in\mathscr{F}}\alpha_{f}\geq d$. ∎

The next result provides a procedure with which to modify a scheme to make it
hyperplane spanning without increasing the complexity exponent:

######  Lemma 2.4.

Suppose that $\mathcal{S}$ is not hyperplane spanning, so there exists some
$H\in\mathscr{H}$ with $\beta_{H}\neq n-1$. Then we may replace $H$ with some
hyperplane $H^{\prime}$, obtaining a new scheme $\mathcal{S}^{\prime}$ with
supporting hyperplanes
$\mathscr{H}^{\prime}=(\mathscr{H}\setminus\\{H\\})\cup\\{H^{\prime}\\}$, so
that:

  1. (1)

$\Gamma^{H}<\Gamma^{H^{\prime}}$;

  2. (2)

each $\alpha_{f}$ satisfies $\alpha_{f}\geq\alpha_{f^{\prime}}$;

  3. (3)

$\beta_{H^{\prime}}=\beta_{H}+1$.

Here, for a flag $f\subset\mathscr{H}$, the flag
$f^{\prime}\subset\mathscr{H}^{\prime}$ has the same hyperplanes as $f$ except
that $H$ is replaced by $H^{\prime}$.

###### Proof.

Let $X\coloneqq\langle\Gamma^{H}\rangle_{\mathbb{R}}$. By assumption,
$\dim(X)=\beta_{H}<n-1$. Choose $x\in H$ and some $\gamma\in\Gamma_{<}$ not in
$X$ which is ‘close’ to $V(H)=H-x$ (which we make precise later). The subspace
$Y\coloneqq\langle X,\gamma\rangle_{\mathbb{R}}$ is of dimension
$\beta_{H}+1$. Let $\Gamma_{<}^{Y}\coloneqq\Gamma_{<}\cap Y$. Then
$\operatorname{rk}(\Gamma_{<}^{Y})>\operatorname{rk}(\Gamma_{<}^{H})$, since
$\Gamma_{<}^{Y}$ contains all lattice elements of
$\Gamma_{<}^{H}=\Gamma_{<}\cap(H-x)$, in addition to the ones coming from the
inclusion of $\gamma$. Choose vectors $v_{1}$, $v_{2}$, …, $v_{\ell}$, with
$\ell=(n-1)-(\beta_{H}+1)$, which complete $Y$ to a subspace

$V\coloneqq\langle Y,v_{1},v_{2},\ldots v_{\ell}\rangle_{\mathbb{R}}$

which is of dimension $n-1$ and contains no lattice points other than those
already contained in $Y$ (since $\Gamma_{<}$ is countable, there is a dense
set of choices for the $v_{i}$). We let $H^{\prime}\coloneqq V+x$. Since
$\Gamma_{<}$ is dense in $\operatorname{\mathbb{E}_{<}}$, we may choose
$\gamma$, $v_{1}$, …, $v_{\ell}$ so that the above conditions hold, and so
that $H^{\prime}$ is ‘close’ to $H$, in the sense that $\mathscr{H}^{\prime}$
is still the set of supporting hyperplanes of some new window, and so that
flags of $\mathscr{H}$ are the same as flags in $\mathscr{H}^{\prime}$, up to
swapping $H$ with $H^{\prime}$ (since being a flag is an open condition on the
set of hyperplanes).

By construction, $\Gamma^{H}<\Gamma^{Y}=\Gamma^{H^{\prime}}$ and
$\beta_{H^{\prime}}=\beta_{H}+1$. Moreover, since
$\Gamma_{<}^{H^{\prime}}\geqslant\langle\Gamma_{<}^{H},\gamma\rangle_{\mathbb{R}}$,
we see that $\operatorname{rk}(H^{\prime})\geq\operatorname{rk}(H)+1$. Hence,
by the definition of $\alpha_{f}$ in Theorem 2.2, for any flag
$f\subset\mathscr{H}$ it is the case that $\alpha_{f}\geq\alpha_{f^{\prime}}$,
since any appearance of $H^{\prime}$ which increases the corresponding
$\beta_{H}$ by $1$ is at least offset by replacing $\operatorname{rk}(H)$ by
$\operatorname{rk}(H^{\prime})$. ∎

Together with Lemma 2.3, the above establishes the following:

######  Corollary 2.5.

Each $\alpha_{f}\geq d$ in Theorem 2.2. In particular, the complexity exponent
$\alpha\geq d$.

###### Proof.

Iteratively apply Lemma 2.4 until $\beta_{H}=n-1$ for each hyperplane, which
does not increase any $\alpha_{f}$. In this way we can always find a new
scheme $\mathcal{S^{\prime}}$ which is hyperplane spanning and has complexity
exponents no higher than the original scheme. Then the result follows from
Lemma 2.3. ∎

The following simple lemma will be particularly useful for low complexity
schemes, when calculating the ranks of the $\Gamma^{I}$:

######  Lemma 2.6.

For $I$, $J\subseteq\mathscr{H}$ we have that

$\operatorname{rk}(I\cup
J)\geq\operatorname{rk}(I)+\operatorname{rk}(J)-\operatorname{rk}(I\cap J),$

with equality if and only if $\Gamma^{I}+\Gamma^{J}$ is a finite index
subgroup of $\Gamma^{I\cap J}$.

###### Proof.

We have a short exact sequence

$0\to\Gamma^{I}\cap\Gamma^{J}\to\Gamma^{I}\times\Gamma^{J}\to\Gamma^{I}+\Gamma^{J}\to
0,$

where the second arrow is the map $x\mapsto(x,-x)$ and the third is
$(x,y)\mapsto x+y$. By Lemma 1.1 we have that

$\Gamma^{I}\cap\Gamma^{J}=(\Gamma\cap V^{I})\cap(\Gamma\cap
V^{J})=\Gamma\cap(V^{I}\cap V^{J})=\Gamma\cap V^{I\cup J}=\Gamma^{I\cup J}.$

Since
$\operatorname{rk}(\Gamma^{I}\times\Gamma^{J})=\operatorname{rk}(I)+\operatorname{rk}(J)$,
it follows from the above short exact sequence that

$\operatorname{rk}(I\cup
J)=\operatorname{rk}(I)+\operatorname{rk}(J)-\operatorname{rk}(\Gamma^{I}+\Gamma^{J}).$

Any $\gamma\in\Gamma^{I}+\Gamma^{J}$ certainly stabilises any hyperplane which
is in both $I$ and $J$, that is $\Gamma^{I}+\Gamma^{J}\leqslant\Gamma^{I\cap
J}$. In particular,
$\operatorname{rk}(\Gamma^{I}+\Gamma^{J})\leq\operatorname{rk}(I\cap J)$, with
equality if and only if the inclusion is of a finite index subgroup, so the
result follows. ∎

The following lemma shows that, for a hyperplane spanning scheme satisfying C,
the lattice $\Gamma$, up to finite index, splits as a sum of subgroups which
project to $n$ complementary lines lying along the intersections of $n-1$
supporting subspaces of the window. The result below will also be useful
later, so we note here that the hyperplane spanning condition may be dropped
(since it follows from C) once Theorem 2.8 has been proven.

######  Lemma 2.7.

Suppose that $\mathcal{S}$ is hyperplane spanning and satisfies C. Choose any
flag $f=\\{V_{1},V_{2},\ldots,V_{n}\\}\subseteq\mathscr{H}_{0}$ and consider
the subsets $\widehat{i}\subset f$ given by
$\widehat{i}=f\setminus\\{V_{i}\\}$. Then
$\Gamma^{\widehat{1}}+\Gamma^{\widehat{2}}+\cdots+\Gamma^{\widehat{n}}$ is a
finite index subgroup of $\Gamma$. Each $\Gamma^{\widehat{i}}_{<}$ is
contained in the $1$-dimensional subspace $L(i)$, given as the intersection of
the $V\in\widehat{i}$; these lines give a direct sum decomposition of
$\operatorname{\mathbb{E}_{<}}$.

###### Proof.

By Lemma 1.1, we have that $\Gamma^{\widehat{i}}_{<}=\Gamma_{<}\cap L(i)$,
where $L(i)$ is the intersection of subspaces in $\widehat{i}$. The
intersection of any subspace $X$ with a codimension $1$ subspace has dimension
$\dim(X)$ or $\dim(X)-1$, so we see that, as an intersection of $n-1$ such
subspaces, $\dim(L(i))\geq 1$. We must have that $\dim(L(i))\leq 1$, or else
$\dim(L(i)\cap V_{i})>0$, contradicting $f$ being a flag, so each $L(i)$ is
$1$-dimensional. The lines $L(i)$ are complementary. Indeed, suppose that
$v_{1}+\cdots+v_{n}=0$, where each $v_{i}\in L(i)$. Then any given $v_{i}$ may
be expressed as a linear sum of the other $v_{j}$. But each such $v_{j}\in
V_{i}\supseteq L(j)$, which implies that $v_{i}\in V_{i}$. Hence, $v_{i}\in
V_{i}\cap L(i)=\\{0\\}$, since $f$ is a flag, so each $v_{i}=0$, as required.

To see that the $\Gamma^{\widehat{i}}$ have sum which is finite rank in
$\Gamma$, we calculate their ranks using the previous Lemma. We let $r(i)$
denote the rank of $\Gamma^{V_{i}}$ and $r(123\cdots i)$ denote the rank of
$\Gamma^{I}$, with $I=\\{V_{1},V_{2},\ldots,V_{i}\\}$. We calculate

$\displaystyle r(1)+r(2)+\cdots+r(n)$ $\displaystyle\leq
r(12)+r(3)+r(4)+\cdots+r(n)+k$ $\displaystyle\leq r(123)+r(4)+\cdots+r(n)+2k$
$\displaystyle\cdots$ $\displaystyle\leq r(123\cdots(n-1))+r(n)+(n-2)k.$

In each step of the above, we use the fact that

$r(123\cdots i)+r(i+1)\leq
r(123\cdots(i+1))+\operatorname{rk}(\emptyset)=r(123\cdots(i+1))+k$

by taking $I=\\{V_{1},V_{2},\ldots,V_{i}\\}$ and $J=\\{V_{i+1}\\}$ in Lemma
2.6. By definition,
$r(123\cdots(n-1))=\operatorname{rk}(\Gamma^{\widehat{n}})$ and we see that

$\sum_{V\in
f}\operatorname{rk}(V)\leq\operatorname{rk}(\Gamma^{\widehat{n}})+\operatorname{rk}(V_{n})+(n-2)k.$

Analogously, by leaving out any other $i$ rather than $n$ we see that

(2.1) $\sum_{V\in
f}\operatorname{rk}(V)\leq\operatorname{rk}(\Gamma^{\widehat{i}})+\operatorname{rk}(V_{i})+(n-2)k.$

By Theorem 2.2, property C, Corollary 2.5 and hyperplane spanning (each
$\beta_{H}=n-1$),

(2.2) $\sum_{V\in f}(d-\operatorname{rk}(V)+(n-1))=d,$

that is,

$\sum_{V\in f}\operatorname{rk}(V)=n\cdot(d+n-1)-d=n\cdot k-n-d=(n-1)\cdot k.$

Applying this to Equation 2.1, we have

$(n-1)\cdot
k\leq\operatorname{rk}(\Gamma^{\widehat{i}})+\operatorname{rk}(V_{i})+(n-2)\cdot
k,\text{ that is,}\ \operatorname{rk}(\Gamma^{\widehat{i}})\geq
k-\operatorname{rk}(V_{i}).$

Summing over each $i$ and applying Equation 2.2 we obtain

$\sum_{i=1}^{n}\operatorname{rk}(\Gamma^{\widehat{i}})\geq\sum_{V\in
f}(k-\operatorname{rk}(V))=\left(\sum_{V\in
f}d-\operatorname{rk}(V)+(n-1)\right)+n=d+n=k.$

The subgroups $\Gamma^{\widehat{i}}$ must be linearly independent in
$\operatorname{\mathbb{E}}$ (since they map to complementary lines in
$\operatorname{\mathbb{E}_{<}}$). Since their ranks sum to at least
$k=\operatorname{rk}(\Gamma)$, their sum must in fact have rank $k$ and be
finite index in $\Gamma$. ∎

We conclude this section with its main theorem:

######  Theorem 2.8.

If $\mathcal{S}$ satisfies C then it is hyperplane spanning.

###### Proof.

Suppose that $\mathcal{S}$ is not hyperplane spanning but satisfies C. By
Corollary 2.5, each $\alpha_{f}=d$. Iteratively applying Lemma 2.4, we may
increase each $\beta_{H}$ whilst preserving the property that each
$\alpha_{f}=d$. We do this for all except one hyperplane $K\in\mathscr{H}$,
which we adjust until $\beta_{K}=n-2$. In other words, this scheme (which we
now refer to as $\mathcal{S}$) is ‘one move’ from a hyperplane spanning
scheme; applying Lemma 2.4 one more time, we may replace $K$ for some
hyperplane $K^{\prime}$ to obtain a hyperplane spanning scheme satisfying C,
denoted $\mathcal{S}^{\prime}$.

Take a flag $f$ for the scheme $\mathcal{S}^{\prime}$ which contains
$K^{\prime}$. By C and Corollary 2.5,

(2.3) $\alpha_{f}=\sum_{H\in f}d-\operatorname{rk}(H)+\beta_{H}=d,$

where each $\beta_{H}=n-1$ since $\mathcal{S}^{\prime}$ is hyperplane
spanning.

By the previous Lemma 2.7, $\Gamma$ is the sum (up to finite index) of
subgroups $\Gamma^{\widehat{i}}$, which project to complementary lines defined
by $f$ in the internal space. Consider the subspaces $V$ and $V^{\prime}$,
given by translating $K$ and $K^{\prime}$ over the origin, respectively. Their
intersection $X=V\cap V^{\prime}$ is an $(n-2)$-dimensional linear subspace of
$V$ and contains all points of $\Gamma^{K}_{<}$. At least one line
$L(i)\subseteq V^{\prime}$, containing some $\Gamma^{\widehat{i}}$, is not
contained in $X$ (since otherwise $K$ and $K^{\prime}$ would be parallel and
have the same stabilisers). It follows that, when replacing $K$ with
$K^{\prime}$, a subgroup of rank at least
$\operatorname{rk}(\Gamma^{\widehat{i}})$ is removed, which is linearly
independent from $\Gamma^{K}$.

So the claim follows if each $\operatorname{rk}(\Gamma^{\widehat{i}})\geq 2$.
Indeed, in that case,
$\operatorname{rk}(K^{\prime})\geq\operatorname{rk}(K)+2$, whereas
$\beta_{K^{\prime}}=\beta_{K}+1$, so the complexity exponent $\alpha_{f}$ in
Equation 2.3 increases by at least $1$ when replacing $K^{\prime}$ with $K$,
hence the complexity exponent for $\mathcal{S}$ is strictly greater than $d$,
contradicting it satisfying C.

To see that each $\Gamma^{\widehat{i}}$ has rank at least $2$, note first that
$\Gamma_{<}$ is dense in $\operatorname{\mathbb{E}_{<}}$. If
$\Gamma^{\widehat{i}}$ was only rank $0$ or $1$, then it would be a discrete
subgroup of $\operatorname{\mathbb{E}_{<}}$. The sum of the remaining
$\Gamma^{\widehat{j}}$, for $j\neq i$, all lie inside the hyperplane $V_{i}$
which is complementary to the line $L(i)\supset\Gamma^{\widehat{i}}$. Hence,
the sum group $S=\Gamma^{\widehat{1}}+\cdots+\Gamma^{\widehat{n}}$ is
contained in a discrete union of parallel codimension $1$ hyperplanes of
$\operatorname{\mathbb{E}_{<}}$, so is not dense. Since $S$ is finite index in
$\Gamma$, we have that $\Gamma\leqslant(1/N)\cdot S$ for some
$N\in\mathbb{N}$. Since $(1/N)\cdot S$ is still not dense in
$\operatorname{\mathbb{E}_{<}}$, this contradicts density of $\Gamma$. ∎

##  3\. Decomposable cut and project schemes

One difficulty in treating the very general class of cut and project schemes
considered here is that some schemes may have internal systems which ‘split’
into summands. In this case, the different components can behave like
independent schemes with varying dimensions, meaning that the relevant (and
optimal) Diophantine exponents can vary between factors:

######  Example 3.1.

Consider two codimension $1$, canonical cut and project schemes which are LR,
with total dimensions $k_{1}<k_{2}$. This means that each window is an
interval, with length equal to the length of a projected lattice point, and
the projection of the lattice points to the internal space are each
Diophantine, meaning that non-trivial lattice points of norm at most $r$ in
the total space project to points which are at least distance
$cr^{-(k_{i}-1)}$ from the origin in the internal space. By taking the direct
sums of the physical, total and internal spaces, products of the lattices and
of the windows, we obtain a codimension $2$ cut and project scheme, which can
be shown to be LR by applying our main theorem; indeed, the resulting patterns
are products of two LR patterns. However, the projected lattice $\Gamma_{<}$
(of rank $k_{1}+k_{2}$) is a sum $\Gamma_{<}=T+S$, where
$\operatorname{rk}(S)=k_{1}$, $\operatorname{rk}(T)=k_{2}$ and $S$, $T$ are
contained in complementary $1$-dimensional subspaces. Since $k_{1}<k_{2}$, the
acceptance domains will be typically rectangular and wide in the first
coordinate, but relatively short in the second coordinate. Whilst the
projected lattice as a whole does not satisfy a natural Diophantine condition,
it splits into two factors which are each Diophantine.

To address the irregularities of examples such as the above, we develop a
theory of decompositions of polytopal cut and project schemes. We start by
defining what it means to decompose the window; we turn to the problem of
compatibly splitting the lattice in Section 3.2.

###  3.1. Decomposing windows

Our window will be _decomposable_ if it can be expressed as a Minkowski sum.
The following more abstract definition, however, will be easier to work with
in what is to follow. It will be seen to be equivalent to a geometric product
decomposition of the window in Theorem 3.18. It is not hard to see that the
following definition is equivalent to the definition of decomposability of
Construction 3.5 in [FHK02] when applied to the normals of supporting
hyperplanes of a window:

######  Definition 3.2.

We call the window $W$ decomposable if there exists a decomposition, a
partition $\mathscr{H}_{0}=A\cup B$, with $A$, $B\neq\emptyset$, for which
$\operatorname{\mathbb{E}_{<}}=X_{A}+X_{B}$, where $X_{A}=\bigcap_{V\in A}V$
and $X_{B}=\bigcap_{V\in B}V$. Call $W$ indecomposable if it does not permit a
decomposition.

Since $W$ is a polytope (it is compact) it has vertices, which implies that
$\bigcap_{V\in\mathscr{H}_{0}}V=\\{0\\}$. So if $W$ is decomposable we have
that $\operatorname{\mathbb{E}_{<}}=X_{A}+X_{B}$ gives a direct sum
decomposition of $\operatorname{\mathbb{E}_{<}}$ into complementary subspaces
$X_{A}$ and $X_{B}$, with each $V\in\mathscr{H}_{0}$ containing either $X_{A}$
or $X_{B}$. One way to visualise the above definition is as follows: we can
split $\operatorname{\mathbb{E}_{<}}$ non-trivially as a sum $X_{A}+X_{B}$, so
that every supporting subspace contains either $X_{A}$ or $X_{B}$.

######  Example 3.3.

In codimension $n=1$ there is only one possible supporting subspace for $W$,
the trivial subspace $\\{0\\}$. Hence $\\#\mathscr{H}_{0}=1$, so the scheme is
automatically indecomposable.

######  Example 3.4.

For codimension $n=2$, we have that $\\#\mathscr{H}_{0}\geq 2$ (there are at
least $2$ non-parallel supporting edges). If
$\mathscr{H}_{0}=\\{V_{1},V_{2}\\}$, then $A=\\{V_{1}\\}$, $B=\\{V_{2}\\}$
gives a decomposition of $W$. Note that in this case, $W$ is a parallelogram,
which is a product of intervals. If $\\#\mathscr{H}_{0}>2$, then $W$ is
indecomposable. Indeed, take any decomposition $\mathscr{H}_{0}=A\cup B$ and
assume (without loss of generality) that $\\#A\geq 2$. Two distinct
$1$-dimensional subspaces intersect to the origin, so that $\dim(X_{A})=0$.
Since $\dim(X_{B})\leq 1$ (being contained in any $V\in B$), we cannot have
that $\operatorname{\mathbb{E}_{<}}=X_{A}+X_{B}$.

######  Example 3.5.

In the codimension $n=3$ case, for any valid decomposition
$\mathscr{H}_{0}=A\cup B$, we have that $\dim(X_{A})=2$ and $\dim(X_{B})=1$
(or vice versa). Indeed, as noted, $X_{A}+X_{B}$ gives a direct sum
decomposition of $\operatorname{\mathbb{E}_{<}}$, and $\dim(X_{A})\leq 2$
(since $X_{A}\leqslant V$ for any $V\in A$); similarly, $\dim(X_{B})\leq 2$.
Since (without loss of generality) $\dim(X_{A})=2$, we must have that
$A=\\{V\\}$ for a single $V\in\mathscr{H}_{0}$ (since if it contained any
other supporting hyperplane, the intersection would have strictly smaller
dimension). Each element of $B$ contains the line $X_{B}$. It follows that $W$
has a ‘top’ and ‘bottom’ face, parallel with $V$, and that all other faces are
rotates of each other around the line $X_{B}$, intersecting in edges parallel
to $X_{B}$. So $W$ is a product of a $2$-dimensional polytope and an interval.
If $W$ can not be expressed as such a product, then it is indecomposable.

We now give an equivalent definition of indecomposability by constructing a
finite, undirected graph using the structure of flags on $\mathscr{H}_{0}$:

######  Definition 3.6.

Call $f\subset\mathscr{H}_{0}$ a pre-flag if there exists some
$V\in\mathscr{H}_{0}$ so that $f\cup\\{V\\}$ is a flag. We define the
undirected graph $G(W)$ by setting $\mathscr{H}_{0}$ as the nodes, and connect
$V_{1}$ to $V_{2}$ with an edge if there is a pre-flag $f$ so that
$f\cup\\{V_{i}\\}$ is a flag for both $i=1$ and $i=2$, in which case we write
$V_{1}\sim V_{2}$.

The above is distinct but related to the graphs defined in Construction 3.5 in
[FHK02], which always have only $n$ vertices. The graph $G(W)$ has the
advantage of not depending on a choice of basis (or inner product on the
internal space). It also allows for a more elementary proof of Theorem 3.8
later.

######  Theorem 3.7.

A partition $\mathscr{H}_{0}=A\cup B$ provides a decomposition of $W$ if and
only if $A$ and $B$ are disconnected in $G(W)$.

###### Proof.

Suppose that $\mathscr{H}_{0}=A\cup B$ is a decomposition for $W$. We claim
that for any flag $f=\\{V_{1},V_{2},\ldots,V_{n}\\}$, the set $f_{A}\subseteq
f$ of hyperplanes in $A$ has $\dim(V_{B})$ elements, and analogously for $B$.
Indeed, since $f$ is a flag the sequence $\dim(V_{1})$, $\dim(V_{1}\cap
V_{2})$, …, $\dim(V_{1}\cap V_{2}\cap\cdots,\cap V_{n})$ reduces by $1$ at
each step. Arrange $f$ so that the elements of $f_{A}$ appear first in the
list. Then we see that $\dim(X_{A})\leq n-\\#f_{A}$, since $X_{A}$ is
contained in the intersection of elements of $f_{A}$, and analogously
$\dim(X_{B})\leq n-\\#f_{B}$. Hence $\\#f_{A}=n-\\#f_{B}\geq\dim(X_{B})$, and
similarly $\\#f_{B}\geq\dim(X_{A})$. Neither of these inequalities can be
strict, since otherwise $n=\\#f_{A}+\\#f_{B}>\dim(V_{B})+\dim(X_{A})=n$, a
contradiction, so the claim follows.

It is now clear that $A$ and $B$ are disconnected in $G(W)$. Indeed, suppose
that $V_{a}\in A$, $V_{b}\in B$ and $V_{a}\sim V_{b}$, so that there is a pre-
flag $f$ which is completed to a flag by adding either $V_{a}$ or $V_{b}$. But
adjoining $V_{a}$ gives a flag with one more element in $A$ than by adjoining
$V_{b}$, contradicting that the numbers $\\#f_{A}$ and $\\#f_{B}$ only depend
on $\dim(X_{A})$ and $\dim(X_{B})$.

For the converse direction, suppose that $A$ and $B$ are disconnected in
$G(W)$. We need to show that $\operatorname{\mathbb{E}_{<}}=X_{A}+X_{B}$. Take
any flag
$f=\\{V_{a_{1}},V_{a_{2}},\ldots,V_{a_{s}},V_{b_{1}},\ldots,V_{b_{t}}\\}$,
where each $V_{a_{i}}\in A$ and $V_{b_{i}}\in B$. Let $X^{\prime}_{A}=\bigcap
V_{a_{i}}$ and $X^{\prime}_{B}=\bigcap V_{b_{i}}$ (that is, we only intersect
the elements of the flag here, rather than all elements of $A$ for $X_{A}$ and
all elements of $B$ for $X_{B}$). Clearly $X_{A}^{\prime}\supseteq X_{A}$,
$X_{B}^{\prime}\supseteq X_{B}$ and
$\operatorname{\mathbb{E}_{<}}=X_{A}^{\prime}+X_{B}^{\prime}$ (since each
intersect trivially, and their dimensions sum to $n$, as $f$ is a flag). We
claim that $X^{\prime}_{A}\subseteq V_{a}$ for all $V_{a}\in A$, and similarly
for $B$, from which the result follows since this implies the reverse
inequalities $X_{A}\subseteq X_{A}^{\prime}$ and $X_{B}\subseteq
X_{B}^{\prime}$.

To see this, let $V_{a}\in A$. If $V_{a}\in f$ then clearly
$X^{\prime}_{A}\subseteq V_{a}$. So suppose not, and consider the set
$f\cup\\{V_{a}\\}$, where we order the elements so that the elements of $A$
appear first, then $V_{a}$, then the elements of $B$:

$f^{\prime}=\\{V_{a_{1}},V_{a_{2}},\ldots,V_{a_{s}},V_{a},V_{b_{1}},V_{b_{2}},\ldots,V_{b_{t}}\\}.$

Since each element of the above is a codimension 1 subspace, taking
intersections

$V_{a_{1}},\ V_{a_{1}}\cap V_{a_{2}},\ V_{a_{1}}\cap V_{a_{2}}\cap V_{a_{3}},\
\ldots$

from left to right drops the dimension by either $0$ or $1$ at each step. If
$X^{\prime}_{A}\nsubseteq V_{a}$ then the dimension drops by $1$ for each new
intersection with an element of $A$, including $V_{a}$. Since there are $n+1$
elements, the dimension must not drop when intersecting with some $V_{b}\in
B\cap f$, that is, some $V_{b}$ contains the intersection of the elements
before it in $f^{\prime}$, so omitting it still results in a set with trivial
intersection, which is thus a flag. So the pre-flag
$f^{\prime}\setminus\\{V_{a},V_{b}\\}$ can be completed to a flag by adjoining
either $V_{a}$ or $V_{b}$, that is, $V_{a}\sim V_{b}$, contradicting that $A$
and $B$ are disconnected in $G(W)$. Hence, $X^{\prime}_{A}\subseteq V_{a}$ for
all $V_{a}\in A$ and analogously for $B$, as required. ∎

The advantage of the definition of a decomposition in terms of the graph of
flags is it allows a simpler proof that minimal complexity schemes with
indecomposable windows have hyperplanes of constant stabiliser rank:

######  Theorem 3.8.

Suppose that $\mathcal{S}$ satisfies C and that $V_{1}$ and $V_{2}$ are
connected by an edge in $G(W)$. Then
$\operatorname{rk}(V_{1})=\operatorname{rk}(V_{2})$.

###### Proof.

By Theorem 2.2 and Corollary 2.5, for any flag $f\subseteq\mathscr{H}_{0}$ we
have

$\sum_{V\in f}(k-\operatorname{rk}(H))=d+n=k,$

since each $\beta_{H}=n-1$ by hyperplane spanning, which follows from C by
Theorem 2.8. It follows that $\sum\operatorname{rk}(H)=k(n-1)$ for each flag.

Suppose then that $V_{1}\sim V_{2}$ in $G(W)$. Take a pre-flag $f$ which is
completed to a flag by adding either $V_{1}$ or $V_{2}$. Then
$\operatorname{rk}(V_{1})=k(n-1)-\left(\sum_{V\in
f}\operatorname{rk}(V)\right)=\operatorname{rk}(V_{2})$, as required. ∎

######  Corollary 3.9.

Suppose that $\mathcal{S}$ satisfies C. If
$\operatorname{rk}(H_{1})\neq\operatorname{rk}(H_{2})$ for some $H_{1}$,
$H_{2}\in\mathscr{H}$, then $W$ is decomposable.

###### Proof.

By the previous theorem, $H_{1}$ and $H_{2}$ belong to different path-
components in $G(W)$, so $G(W)$ is disconnected. Hence $W$ is decomposable by
Theorem 3.7. ∎

The above result allows us to decompose a minimal complexity cut and project
scheme whenever it has hyperplanes with stabilisers of different ranks. This
can be repeated until an indecomposable setup is reached. To make this
concrete, we now give an alternative, more direct description of a
decomposition into more than two pieces. It will turn out that this is
equivalent to iteratively breaking summands into two, as in the above
definitions, and will eventually describe the window as a product of lower
dimensional ones.

######  Definition 3.10.

A decomposition of $W$ is a set $\\{X_{1},X_{2},\ldots,X_{m}\\}$ of $m\geq 2$
subspaces of $\operatorname{\mathbb{E}_{<}}$ for which

  1. (1)

each $X_{i}$ is non-trivial (i.e., not equal to $\\{0\\}$ or
$\operatorname{\mathbb{E}_{<}}$);

  2. (2)

the $X_{i}$ are complementary and give a direct sum decomposition of
$\operatorname{\mathbb{E}_{<}}$;

  3. (3)

for each $V\in\mathscr{H}_{0}$ we have that $V\supseteq X_{i}$ for all but one
$X_{i}$.

We give an alternative version of the above definition, which for $m=2$
clearly extends our original Definition 3.2.

######  Definition 3.11.

For subsets $A_{i}\subseteq\mathscr{H}_{0}$, call $\\{A_{i}\\}_{i=1}^{m}$ a
copartition of $\mathscr{H}_{0}$ if $\\{\mathscr{H}_{0}\setminus
A_{i}\\}_{i=1}^{m}$ is a partition of $\mathscr{H}_{0}$. Call a copartition
non-trivial if each $A_{i}\neq\emptyset$, $\mathscr{H}_{0}$. Then we define a
decomposition of the window $W$ to be a non-trivial copartition
$\\{A_{i}\\}_{i=1}^{m}$ of $\mathscr{H}_{0}$ such that $m\geq 2$ and
$\operatorname{\mathbb{E}_{<}}=X_{1}+\cdots+X_{m}$, where

(3.1) $X_{i}\coloneqq\bigcap_{V\in A_{i}}V.$

Of course, $\\{A_{i}\\}_{i=1}^{m}$ is a co-partition precisely when each
$V\in\mathscr{H}_{0}$ appears in all but one $A_{i}$. The above definition
agrees with Definition 3.2 in the case that $m=2$. We prove below that both
versions, Definition 3.10 and 3.11, are equivalent:

######  Proposition 3.12.

Suppose that $\\{X_{i}\\}_{i=1}^{m}$ is a decomposition of $W$, in the sense
of Definition 3.10. Define $A_{i}$ to be the set of $V\in\mathscr{H}_{0}$ for
which $V\supseteq X_{i}$. Then $\\{A_{i}\\}_{i=1}^{m}$ is a decomposition of
$W$, in the sense of Definition 3.11.

Conversely, suppose that $\\{A_{i}\\}_{i=1}^{m}$ is a decomposition, in the
sense of Definition 3.11. Then the subspaces $\\{X_{i}\\}_{i=1}^{m}$, as
defined in Equation 3.1, are a decomposition of $W$ in the sense of Definition
3.10.

###### Proof.

Suppose that $\\{X_{i}\\}_{i=1}^{m}$ is a decomposition and let $A_{i}$ be as
defined in the statement of the proposition. Clearly $\\{A_{i}\\}_{i=1}^{m}$
is a copartition, since by Definition 3.10 (3) each $V\in\mathscr{H}_{0}$
belongs to all but precisely one $X_{i}$. No $A_{i}=\emptyset$ or
$\mathscr{H}_{0}$ or else $X_{i}=\operatorname{\mathbb{E}_{<}}$ or $\\{0\\}$,
respectively, violating (1). Moreover, the original subspaces $X_{i}$ are at
least contained in those of the intersections in Equation 3.1, so these
intersections sum to $\operatorname{\mathbb{E}_{<}}$ by (2), as required.

For the converse direction, suppose that $\\{A_{i}\\}_{i=1}^{m}$ gives a
decomposition in the sense of Definition 3.11. We must show that properties
(1–3) of Definition 3.10 hold for the $X_{i}$ given by Equation 3.1.

For condition (1), clearly each $X_{i}\neq\operatorname{\mathbb{E}_{<}}$,
since $X_{i}\subseteq V$ for any $V\in A_{i}$ (and each $A_{i}\neq\emptyset$
by assumption). To see that each $X_{i}\neq\\{0\\}$, firstly note that each
$X_{i}\cap X_{j}=\\{0\\}$ for distinct $i$, $j$. Indeed, $A_{i}\cup
A_{j}=\mathscr{H}_{0}$, since any given $V\in\mathscr{H}_{0}$ belongs to at
least one of $A_{i}$ or $A_{j}$. Hence,

$X_{i}\cap X_{j}=\bigcap_{V\in A_{i}\cup
A_{j}}V=\bigcap_{V\in\mathscr{H}_{0}}V=\\{0\\},$

so the $X_{i}$ intersect trivially. By assumption,
$X_{i}+\hat{X}_{i}=\operatorname{\mathbb{E}_{<}}$, where $\widehat{X}_{i}$ is
the direct sum of all $X_{j}$ with $j\neq i$. Then $X_{i}\neq\\{0\\}$ follows
from $\widehat{X}_{i}\neq\operatorname{\mathbb{E}_{<}}$, which holds since
each $X_{j}\subseteq V$, for $j\neq i$, where $V\in\mathscr{H}_{0}$ is any
element not contained in $A_{i}$ (and thus is contained in every other
$A_{j}$); such an element exists since by assumption each
$A_{i}\neq\mathscr{H}_{0}$.

The above argument shows that the $X_{i}$ intersect trivially, and by
assumption their sum spans $\operatorname{\mathbb{E}_{<}}$, so (2) is
satisfied. Finally, by Equation 3.1, $V\supseteq X_{i}$ if $V\in A_{i}$ (since
$X_{i}$ is the intersection of such $V$), and $V\in A_{i}$ for all but one
$i$, since $\\{A_{i}\\}_{i=1}^{m}$ is a copartition. For the unique value of
$i$ with $V\notin X_{i}$, we cannot have $V\supseteq X_{i}$, since otherwise
the sum of the $X_{i}$ would be contained in $V$, contradicting the sum
spanning $\operatorname{\mathbb{E}_{<}}$, so (3) holds. ∎

By the above, we may use either of the two definitions of a decomposition (in
terms of the $X_{i}$ or $A_{i}$) interchangeably. The following corollary of
the above will be useful when analysing how the lattice decomposes with
respect to a decomposition, and also in showing that any two decompositions
have a common refinement:

######  Corollary 3.13.

If $\\{X_{i}\\}_{i=1}^{m}$ is a decomposition of $W$, then each $X_{i}$ is
equal to the intersection of hyperplanes of $\mathscr{H}_{0}$ containing it.

###### Proof.

By the above proposition, the subspaces $X_{i}^{\prime}$ given by intersecting
the $V\in\mathscr{H}_{0}$ containing some $X_{i}$ give a decomposition in the
sense of Definition 3.10. Clearly each $X_{i}\subseteq X_{i}^{\prime}$, but
none of these can be a strict inclusion or else the $X_{i}^{\prime}$ would not
be complementary, violating Definition 3.10 (2). ∎

######  Proposition 3.14.

Suppose that $\\{X_{i}\\}_{i=1}^{m}$ is a decomposition, with $m\geq 3$. Then
replacing the terms $X_{i}$, $X_{j}$ ($i\neq j$) with $X_{i}+X_{j}$ gives
another decomposition of $m-1$ elements. In terms of Definition 3.11, for the
copartition $\\{A_{i}\\}_{i=1}^{m}$ this corresponds to replacing $A_{i}$ and
$A_{j}$ with $A_{i}\cap A_{j}$.

###### Proof.

Conditions (1–2) of Definition 3.10 clearly still hold after replacing $X_{i}$
and $X_{j}$ with their sum. For each $V\in\mathscr{H}_{0}$, if $V\supseteq
X_{i}$, $X_{j}$, then $V\supseteq X_{i}+X_{j}$ (since $V$ is a vector
subspace). Hence, if $X_{\ell}$ with $\ell\neq i$, $j$ is the unique subspace
which is not contained in $V$, this property still holds in the new
decomposition. Otherwise, if $\ell=i$, then $X_{i}+X_{j}\nsubseteq V$, or else
$X_{i}\subseteq V-X_{j}\subseteq V-V=V$, a contradiction. Similarly, if
$\ell=j$ then $X_{i}+X_{j}\nsubseteq V$, so (3) is satisfied and we have a
decomposition.

In terms of coparitions, we first define the copartition
$\\{A_{i}\\}_{i=1}^{m}$ as in Proposition 3.12. Then for $\ell\neq i$, $j$,
$A_{i}$ is the set of $V\in\mathscr{H}_{0}$ for which $V\supseteq X_{i}$. On
the other hand, $A_{i}\cap A_{j}$ is precisely the subset of hyperplanes for
which $V\supseteq X_{i}+X_{j}$. Indeed, if $V\in A_{i}\cap A_{j}$, then by
definition $V\supseteq X_{i}$ and $V\supseteq X_{j}$, so $V\supseteq
X_{i}+X_{j}$, since $V$ is a subspace. On the other hand, if $V\notin A_{i}$
or $V\notin A_{j}$, then $V\nsupseteq X_{i}+X_{j}$, as above. So replacing
$A_{i}$ and $A_{j}$ with $A_{i}\cap A_{j}$ corresponds to replacing $X_{i}$
and $X_{j}$ with $X_{i}+X_{j}$, according to Proposition 3.12. ∎

Recall that for $m=2$, Definition 3.11 of a decomposition in terms of
copartitions coincides with Definition 3.2. The above shows that we may define
a window to be decomposable if it has a decomposition, using any of the three
previous definitions.

The following result generalises an argument from the proof of Theorem 3.7.
The first part of it will be used later in showing that decompositions have
common refinements:

######  Proposition 3.15.

Let $\\{X_{i}\\}_{i=1}^{m}$ be a decomposition of $W$ and $f$ be a flag in
$\mathscr{H}_{0}$. Define $X_{i}^{f}$ to be the intersection of hyperplanes
$V\in f$ which contain $X_{i}$. Then $X_{i}^{f}=X_{i}$ for each $i$. Moreover,
the number of elements in any flag $f$ which contain $X_{i}$ is given by
$n-\dim(X_{i})$. That is, $\\#(f\cap A_{i})=n-\dim(X_{i})$ for each $i$, where
$\\{A_{i}\\}_{i=1}^{m}$ is the equivalent copartition description of the
decomposition coming from Proposition 3.12.

###### Proof.

We claim that $X_{i}^{f}\cap X_{j}^{f}=\\{0\\}$ for distinct $i$ and $j$.
Indeed, given any $V\in f$ we have that $V\supseteq X_{i}$ or $V\supseteq
X_{j}$, by (3) of Definition 3.10. Hence $V\supseteq X_{i}^{f}\cap X_{j}^{f}$.
Since this holds for all elements of $f$, and the elements of a flag intersect
to $\\{0\\}$, we must have that $X_{i}^{f}\cap X_{j}^{f}=\\{0\\}$.

Clearly $X_{i}^{f}\supseteq X_{i}$ for each $i$. Since the $X_{i}^{f}$
intersect trivially, and the sum of $X_{i}$ span
$\operatorname{\mathbb{E}_{<}}$, for dimensional reasons we must have that
$X_{i}^{f}=X_{i}$ for each $i$.

By definition,

$X_{i}^{f}=\bigcap_{V\in f\cap A_{i}}V,$

where $A_{i}$ is defined as the set of $V\in\mathscr{H}_{0}$ containing
$X_{i}$, as in Proposition 3.12. Since $f$ is a flag, the dimension of the
above intersection is given by $\dim(X_{i}^{f})=n-\\#(f\cap A_{i})$. Since
$X_{i}=X_{i}^{f}$, the result follows. ∎

We now turn to the problem of relating this abstract and combinatorial
definition of a decomposition to a more geometric one: expressing the window
as a Minkowski sum. We first introduce some important subspaces associated to
a decomposition which define the factors of the window.

######  Notation 3.16.

Given a decomposition $\\{X_{1},\ldots,X_{m}\\}$ of $W$ and
$i\in\\{1,\ldots,m\\}$:

  1. (1)

$\widehat{X}_{i}$ is defined as the sum of all of the $X_{j}$ for $j\neq i$;

  2. (2)

for $H\in\mathscr{H}$ we let $X_{H}$ be the unique $X_{i}$ for which $V(H)$
does not contain $X_{i}$, and write $\widehat{X}_{H}$ for $\widehat{X}_{i}$

  3. (3)

$H_{X}\coloneqq H\cap X_{H}$;

  4. (4)

$H^{+}_{X}\coloneqq H^{+}\cap X_{H}$;

  5. (5)

$V_{X}=V\cap X_{H}$ for $V=V(H)$.

We note that $H_{X}$, $H_{X}^{+}$ and $V_{X}$ are an affine hyperplane, half
space and codimension $1$ subspace in $X_{H}$, respectively (since
$X_{H}\nsubseteq V(H)$), which aids in remembering this notation (in line with
how we use $H$, $H^{+}$ and $V$). Terms $X$ and $\widehat{X}$ (with
subscripts) are complementary subspaces.

######  Lemma 3.17.

If $\\{X_{1},\ldots,X_{m}\\}$ is a decomposition of $W$, then for
$H\in\mathscr{H}$ and $V=V(H)$ we have:

  1. (1)

$H=H_{X}+\widehat{X}_{H}$;

  2. (2)

$H^{+}=H_{X}^{+}+\widehat{X}_{H}$;

  3. (3)

$V=V_{X}+\widehat{X}_{H}$.

###### Proof.

We start with the third item. Since $V$ is codimension $1$ we have that
$V_{X}$ has dimension $\dim(X_{H})-1$, indeed, $X_{H}$ is not contained in $V$
by the definition of $X_{H}$. So $V_{X}+\widehat{X}_{H}$ has the correct
dimension $n-1$. Clearly if $v\in\widehat{X}_{H}$ then $v\in V$, and if $v\in
V_{X}$ then $v\in V$, so it follows that $V=V_{X}+\widehat{X}_{V}$.

Let $v\in\operatorname{\mathbb{E}_{<}}$ be such that $H+v=V$. Since
$X_{H}+\widehat{X}_{H}$ is a direct sum decomposition of
$\operatorname{\mathbb{E}_{<}}$, we may write $v=v_{H}+\widehat{v}_{H}$, where
$v_{H}\in X_{H}$ and $\widehat{v}_{H}=\widehat{X}_{H}$. Since
$\widehat{X}_{H}\subseteq V$ we may assume that $v=v_{H}$ and
$\widehat{v}_{H}=0$, since
$H=V+v_{H}+\widehat{v}_{H}=(V+\widehat{v}_{H})+v_{H}=V+v_{H}$. Hence
$H=V+v=(V_{X}+v)+\widehat{X}_{H}$, so we need to check that $V_{X}+v=H_{X}$.
Indeed, $V_{X}+v=(V\cap X_{H})+v=(V+v)\cap(X_{H}+v)=(V+v)\cap X_{H}=H\cap
X_{H}=H_{X}$, as required. Item 2 is proved analogously. ∎

Given a decomposition $\\{X_{i}\\}_{i=1}^{m}$ of $W$ and
$i\in\\{1,\ldots,m\\}$, let $\mathscr{H}_{i}$ denote the set of
$H_{X}\subseteq X_{i}$, where $X_{H}=X_{i}$ (that is, $X_{i}\nsubseteq V(H)$),
and $\mathscr{H}_{i}^{+}$ the corresponding half spaces $H_{X}^{+}$. We define
the polytope $W_{i}\subseteq X_{i}$ as:

$W_{i}=\bigcap_{H_{X}^{+}\in\mathscr{H}_{i}}H_{X}^{+}.$

######  Theorem 3.18.

Let $\\{X_{i}\\}_{i=1}^{m}$ be a set of subspaces giving a direct sum
decomposition of $\operatorname{\mathbb{E}_{<}}$. Then $\\{X_{i}\\}_{i=1}^{m}$
is a decomposition of $W$ if and only if there are polytopes $W_{i}\subseteq
X_{i}$ for which $W=W_{1}+\cdots+W_{m}$.

###### Proof.

Suppose that $\\{X_{i}\\}_{i=1}^{m}$ is a decomposition of $W$. By Lemma 3.17,
we have that

$W=\bigcap_{i=1}^{m}\bigcap_{H^{+}\in\mathscr{H}_{i}^{+}}(H^{+}_{X}+\widehat{X}_{H})=\bigcap_{i=1}^{m}\left(\left(\bigcap_{H^{+}\in\mathscr{H}_{i}^{+}}H^{+}_{X}\right)+\widehat{X}_{i}\right)=\bigcap_{i=1}^{m}(W_{i}+\widehat{X}_{i}).$

The second equality above follows from observing that the $\widehat{X}_{H}$
coordinates of the $i$th term of the left-hand intersection must be taken
equal, since each $H_{X}^{+}$ is contained in $X_{H}$, which is complementary
to $\widehat{X}_{H}$.

Given $w=w_{1}+w_{2}+\cdots+w_{m}\in W_{1}+W_{2}+\cdots W_{m}$, clearly $w$ is
in right-hand intersection above, since $w=w_{i}+(w-w_{i})$, and
$w-w_{i}=w_{1}+\cdots+w_{i-1}+w_{i+1}+\cdots+w_{m}\in\widehat{X}_{i}$ for each
$i$. Conversely, given an element $w$ of the above intersection, for each $i$
we may write $w=w_{i}+\widehat{w}_{i}$ for $w_{i}\in W_{i}$ and
$\widehat{w}_{i}\in\widehat{X}_{i}$. Then $w=w_{1}+w_{2}+\cdots w_{m}$, since
this has the same $X_{i}$ components and the $X_{i}$ give a direct sum
decomposition. So $W=W_{1}+\cdots W_{m}$. Since $W$ is compact, the polyhedra
$W_{i}$ must be compact and hence, as intersections of half spaces, are
polytopes.

So now suppose that $W=W_{1}+W_{2}+\cdots+W_{m}$ for polytopes $W_{i}\subseteq
X_{i}$. Let $\mathscr{H}^{+}_{i}$ be the (irredundant) set of half-spaces in
$X_{i}$ with intersection $W_{i}$. We claim that $\mathscr{H}^{+}$ is the set
of half-spaces $H^{+}+\widehat{X}_{i}$ over all $H^{+}\in\mathscr{H}^{+}_{i}$
and $i\in\\{1,\ldots,m\\}$. Indeed, as above, $W$ is the intersection of these
half-spaces. The set $\mathscr{H}^{+}$ is irredundant since omitting any one
element, say $H_{i}+\widehat{X}_{i}$, leads to an intersection equal to
$W_{1}+\cdots+W_{i-1}+W_{i}^{\prime}+W_{i+1}+\cdots W_{m}$, where
$W_{i}^{\prime}$ is strictly larger than $W_{i}$ by the fact that
$\mathscr{H}^{+}_{i}$ is irredundant. Finally, we have that
$V(H+\hat{X}_{i})=(V(H)+\widehat{X}_{i})\supseteq\widehat{X}_{i}\supseteq
X_{j}$ for $H\in\mathscr{H}_{i}$ and $j\neq i$, so each hyperplane of
$\mathscr{H}_{0}$ contains all but one of the $X_{i}$, as required. ∎

The result below shows that we may always decompose uniquely into
indecomposables:

######  Proposition 3.19.

Any two decompositions $X=\\{X_{i}\\}_{i=1}^{q}$ and $Y=\\{Y_{i}\\}_{i=1}^{r}$
of $W$ have a common refinement, that is, a decomposition
$Z=\\{Z_{i}\\}_{i=1}^{s}$ such that each $Z_{i}$ is contained in an element of
$X$ and an element of $Y$.

###### Proof.

Define the decomposition $Z$ to be

$\\{X_{i}\cap Y_{j}\mid X_{i}\in X,Y_{j}\in Y\\},$

omitting elements with trivial intersection, so Definition 3.10 (1) is
satisfied by construction. Clearly $Z$ refines both $X$ and $Y$, and distinct
elements intersect trivially (since distinct elements of both $X$ and $Y$
intersect trivially). Any $V\in\mathscr{H}_{0}$ wholly contains every
$X_{i}\cap Y_{j}$ except for $X_{H}\cap Y_{H}$. It must be that $V\nsupseteq
X_{H}\cap V_{H}$, and hence that (3) holds, once we have shown (2) (that the
sum of $Z_{i}\in Z$ span $\operatorname{\mathbb{E}_{<}}$), since otherwise
each element of $Z$ is contained in $V$, contradicting that they span
$\operatorname{\mathbb{E}_{<}}$.

To prove (2), take any flag $f\subseteq\mathscr{H}_{0}$ and consider the
$1$-dimensional subspace $L(V)$ given by intersecting all elements of $f$
except for $V\in f$. Just as in the proof of Lemma 2.7, these are
$1$-dimensional subspaces which give a direct sum decomposition of
$\operatorname{\mathbb{E}_{<}}$, so it will suffice to show that some element
of $X$ and some element of $Y$ contains $L(V)$.

Recall the $X_{i}^{f}$ from Proposition 3.15. Given $V\in f$, take the unique
$i$ with $V\nsupseteq X_{i}$. By definition, $X_{i}^{f}$ is an intersection of
elements of $f$ not including $V$, so $L(V)$, being the intersection of all
elements of $f$ except $V$, must be a subset of $X_{i}^{f}$. Then
$L(V)\subseteq X_{i}$ since $X_{i}^{f}=X_{i}$ by Proposition 3.15. By the same
argument, there is some $Y_{j}\in Y$ containing $L(V)$, so $L(V)\subseteq
X_{i}\cap Y_{j}$, as required. ∎

######  Corollary 3.20.

Every window $W$ has a unique and minimal decomposition
$\\{X_{i}\\}_{i=1}^{m}$ so that $W=W_{1}+\cdots+W_{m}$, where each
$W_{i}\subseteq X_{i}$ is indecomposable.

In summary, there always exists a decomposition of a window into a sum of
indecomposables. Furthermore, when assuming minimal complexity C, the
supporting hyperplanes in each summand have equal ranks by Corollary 3.9.

###  3.2. Decomposing schemes: Splitting both $\Gamma$ and
$\operatorname{\mathbb{E}_{<}}$

The acceptance domains (or cut regions) are defined not just by the window but
also by how it is aligned relative to the projected lattice. We may not
effectively apply the above techniques without a corresponding decomposition
of $\Gamma$. Fortunately, for schemes satisfying C, the geometric
decomposition of the window gives an associated decomposition of the lattice,
up to finite index:

######  Definition 3.21.

Let $\\{X_{i}\\}_{i=1}^{m}$ be a decomposition of $W$, with associated
copartition decomposition $\\{A_{i}\\}_{i=1}^{m}$. We define
$\Gamma_{i}=\Gamma^{A_{i}}$, the subgroup which stabilises the hyperplanes of
$A_{i}$. Equivalently, $\Gamma_{i}$ is determined by
$(\Gamma_{i})_{<}=\Gamma_{<}\cap X_{i}$.

######  Remark 3.22.

Note that $(\Gamma^{A_{i}})_{<}=\Gamma_{<}\cap X_{i}$ by Lemma 1.1 and the
definition of $X_{i}$ in terms of $A_{i}$ by Definition 3.11, which agrees
with the original decomposition $\\{X_{i}\\}_{i=1}^{m}$ by Proposition 3.12.
This uniquely defines $\Gamma_{i}\leqslant\Gamma$ since $\pi_{<}$ is
injective.

######  Theorem 3.23.

Suppose that $\mathcal{S}$ satisfies C. Then $\Gamma_{1}+\cdots+\Gamma_{m}$ is
finite index in $\Gamma$.

###### Proof.

We shall prove the result by induction on the number of elements of the
decomposition. Recall from Proposition 3.14 that for a decomposition
$\\{A_{i}\\}_{i=1}^{m}$, $m\geq 3$, the decomposition given by replacing the
two elements $A_{i}$ and $A_{j}$ ($i\neq j$) with $A_{i}\cap A_{j}$ is also a
decomposition. So, for $\ell=2$, …, $m$, we have decompositions of $\ell$
elements given by

$D_{\ell}\coloneqq\\{A_{1},A_{2},\ldots,A_{\ell-1},B_{\ell}\\}\ ,\text{ where
}B_{\ell}\coloneqq\bigcap_{j=\ell}^{m}A_{j}.$

Note that $D_{m}=\\{A_{1},\ldots,A_{m}\\}$ is our target decomposition.
Although not strictly a decomposition, we can also define the ‘one element
decomposition’ $D_{1}=\\{\emptyset\\}$, for which the result holds trivially,
as $\Gamma^{\emptyset}=\Gamma$. So suppose that the result holds for the
decompositions $D_{1}$, $D_{2}$, …, $D_{\ell}$. We shall show that it holds
for $D_{\ell+1}$.

We have that $D_{\ell}$ is obtained from $D_{\ell+1}$ by replacing $A_{\ell}$
and $B_{\ell+1}$ by their intersection $B\coloneqq B_{\ell}$. Note that
$A_{\ell}\cup B_{\ell+1}=\mathscr{H}_{0}$ (since every element of
$\mathscr{H}_{0}$ is in at least one of these subsets, by Definition 3.11) so
we may write

$A_{\ell}=B\cup I\ ,\ B_{\ell+1}=B\cup J,$

for the last two elements of $D_{\ell+1}$, where $I$ and
$J\subseteq\mathscr{H}_{0}$ are chosen so that $\mathscr{H}_{0}=B\cup I\cup J$
is a partition. Now, by our induction assumption,

$\Gamma_{1}+\Gamma_{2}+\cdots+\Gamma_{\ell-1}+\Gamma^{B}\leqslant\Gamma$

is a finite index inclusion. So to obtain the corresponding result for
$D_{\ell+1}$, we need that

$\Gamma^{B\cup I}+\Gamma^{B\cup J}\leqslant\Gamma^{B}$

is a finite index inclusion.

Choose any flag $f$ and let $f_{B}\coloneqq f\cap B$, $f_{I}\coloneqq f\cap I$
and $f_{J}\coloneqq f\cap J$. Let $\alpha=\\#f_{I}$ and $\beta=\\#f_{J}$. By
C, Theorem 2.2 and Corollary 2.5,

(3.2) $\sum_{V\in f}\operatorname{rk}(V)=\left(\sum_{V\in
f_{B}}\operatorname{rk}(V)\right)+\left(\sum_{V\in
f_{I}}\operatorname{rk}(V)\right)+\left(\sum_{V\in
f_{J}}\operatorname{rk}(V)\right)=(n-1)\cdot k,$

since $\mathcal{S}$ is hyperplane spanning by Theorem 2.8. By Lemma 2.6, for
any $S\subseteq\mathscr{H}_{0}$ and $V\in S$,

(3.3)
$\operatorname{rk}(S)\geq\operatorname{rk}({S-\\{V\\}})+\operatorname{rk}(V)-\operatorname{rk}(\emptyset)=\operatorname{rk}(S-\\{V\\})+\operatorname{rk}(V)-k.$

Recall from Proposition 3.15 that for any flag $f$, we have that
$X_{i}=X_{i}^{f}$ for a decomposition $\\{X_{i}\\}_{i=1}^{m}$. That is, one
may restrict intersections to any given flag in determining the subspace
intersections $X_{i}$; moreover, the stabiliser subgroups $\Gamma^{A_{i}}$ are
the intersections with these $X_{i}$, once projected to
$\operatorname{\mathbb{E}_{<}}$. In particular, $\operatorname{rk}(B\cup
I)=\operatorname{rk}(B\cup f_{I})$ (once the elements of the flag $f_{I}$ are
stabilised, so too are all elements of $I$). So by inductively applying
Equation 3.3 to successively remove elements of $f_{I}$ from $B\cup f_{I}$, we
see that

$\operatorname{rk}(B\cup I)=\operatorname{rk}(B\cup
f_{I})\geq\operatorname{rk}(B)+\left(\sum_{V\in
f_{I}}\operatorname{rk}(V)\right)-\alpha\cdot k,$

and similarly for $B\cup J$; summing these together, we obtain:

(3.4) $\operatorname{rk}(B\cup I)+\operatorname{rk}(B\cup J)\geq
2\cdot\operatorname{rk}(B)+\left(\sum_{V\in f_{I}\cup
f_{J}}\operatorname{rk}(V)\right)-(\alpha+\beta)\cdot k.$

Similarly for $B=B_{\ell}$,

(3.5) $\operatorname{rk}(B)=\operatorname{rk}(f_{B})\geq\left(\sum_{V\in
f_{B}}\operatorname{rk}(V)\right)-((n-(\alpha+\beta))-1)\cdot k,$

where in this case $f_{B}$ has $n-(\alpha+\beta)$ elements and we need to
apply (3.3) $n-(\alpha+\beta)-1$ times to break it into singletons.

Combining (3.2), (3.4) and (3.5),

$\operatorname{rk}(B\cup I)+\operatorname{rk}(B\cup
J)\geq\operatorname{rk}(B)+\left(\sum_{V\in
f}\operatorname{rk}(V)\right)-(n-1)\cdot k=\operatorname{rk}(B).$

On the other hand, $\Gamma^{B\cup I}$, $\Gamma^{B\cup J}$ have trivial
intersection (since $\mathscr{H}_{0}=B\cup I\cup J$, and no non-trivial
element of $\Gamma$ can stabilise all elements of $\mathscr{H}_{0}$), so the
sum of their ranks is the rank of their sum. Hence, having equal rank,
$\Gamma^{B\cup I}+\Gamma^{B\cup J}$ is finite index in $\Gamma^{B}$, as
required. ∎

Since $\Gamma_{1}+\cdots+\Gamma_{m}$ is finite rank in $\Gamma$, each
$\Gamma_{i}$ projects densely to $X_{i}$ (or else the sum does not project
densely to $\operatorname{\mathbb{E}_{<}}$, contradicting that $\Gamma$ does).
The following corollary summarises the main conclusions of this section and
establishes some further notation.

######  Corollary 3.24.

If $\mathcal{S}$ is aperiodic and satisfies C then it may be split into
components, in the following sense:

  1. (1)

$\operatorname{\mathbb{E}_{<}}=X_{1}+\cdots+X_{m}$ with the subspaces $X_{i}$
complementary;

  2. (2)

$W=W_{1}+\cdots+W_{m}$ where each $W_{i}\subset X_{i}$ is a polytope;

  3. (3)

$\Gamma_{1}+\cdots+\Gamma_{m}$ is a finite index subgroup of $\Gamma$, and
each $\Gamma_{i}$ projects densely into $X_{i}$.

For each $H\in\mathscr{H}$ there is exactly one
$X_{H}\in\\{X_{i}\\}_{i=1}^{m}$ such that $V(H)\not\supseteq X_{H}$. For each
$i=1,\dots,m$, let

$\mathscr{H}_{i}=\\{H\cap X_{i}\mid H\in\mathscr{H},V(H)\nsupseteq X_{i}\\}\
,\ \ \mathscr{H}_{i}^{+}=\\{H^{+}\cap X_{i}\mid
H^{+}\in\mathscr{H}^{+},V(H)\nsupseteq X_{i}\\}.$

Then $H\in\mathscr{H}_{i}$ are affine hyperplanes in $X_{i}$ and give the
sides of the polytope

$W_{i}=\bigcap_{H^{+}\in\mathscr{H}_{i}^{+}}H^{+}.$

For $H\in\mathscr{H}_{i}$, given by $\tilde{H}\cap X_{i}$ for some
$\tilde{H}\in\mathscr{H}$, we have the stabiliser of $H$ in $\Gamma_{i}$:

$\Gamma_{i}^{H}=\Gamma_{i}\cap\Gamma^{\tilde{H}}=\\{\gamma\in\Gamma_{i}\mid\gamma_{<}\in
V(H)\\}.$

The rank $\operatorname{rk}(H)=\operatorname{rk}(\Gamma_{i}^{H})$ will
sometimes be denoted by $\operatorname{rk}_{i}(H)$ if we wish to emphasise
these as coming from subgroups of $\Gamma_{i}$. We may always choose a
sufficiently fine decomposition (for example, by passing to indecomposables)
so that for a fixed $i=1,\dots,m$, the ranks of $\Gamma_{i}^{H}$ for
$H\in\mathscr{H}_{i}$ are all equal. We denote this common rank by $r_{i}$.
This rank is determined by $k_{i}$ and $n_{i}$ when property C is assumed (see
Proposition 4.6).

######  Definition 3.25.

We call each $(X_{i},\Gamma_{i},W_{i})$ a subsystem. We denote
$k_{i}\coloneqq\operatorname{rk}\Gamma^{i}$, $n_{i}\coloneqq\dim X_{i}$ and
$d_{i}\coloneqq k_{i}-n_{i}$. By Corollary 3.24 there is a number $N$ such
that

$\Gamma_{1}+\cdots+\Gamma_{m}\leqslant\Gamma\leqslant\tfrac{1}{N}\Gamma_{1}+\cdots+\tfrac{1}{N}\Gamma_{m}.$

When context makes it clear, we shall sometimes also refer to
$(X_{i},\frac{1}{N}\Gamma_{i},W_{i})$ as a subsystem.

####  3.2.1. Complexity and repetitivity of subsystems

For $r>0$, we let $\Gamma_{i}(r)\coloneqq B_{r}(0)\cap\Gamma_{i}$, where
$B_{r}(0)$ is the closed ball of radius $r$ at the origin in the total space
$\operatorname{\mathbb{E}}$. Most of the constructions and definitions we have
seen need only the data of the subsystems. For example, the stabiliser
subgroups (as discussed above), acceptance domains and cut regions may be
given the same definitions, as well as the hyperplane spanning and (weakly)
homogeneous conditions.

Even though the subsystems do not have canonically associated physical spaces
(although one may somewhat arbitrarily assign a physical dimension and
projection), the associated ‘lifted’ cut and project sets still exist:

$\operatorname{\Lambda}_{i}^{\wedge}\coloneqq\\{\gamma\in\Gamma_{i}\mid\gamma_{<}\in
W_{i}\\}.$

We may define the complexity and repetitivity function in terms of the
acceptance domains (see Notation 4.7). So the complexity function $p(r)$ is
given by the number of distinct $r$-acceptance domains. The repetitivity
function $\rho(r)$ is the smallest value of $R$ so that for every
$\gamma\in\operatorname{\Lambda}_{i}^{\wedge}$, and every
$A\in\mathscr{A}_{i}(r)$, there is some $\gamma^{\prime}\in
B_{R}(\gamma)\cap\Gamma_{i}$ with $\gamma_{<}^{\prime}\in A$. We may thus
define a subsystem to be linearly repetitive if it has repetitivity
$\rho(r)\leq Cr$ for some $C>0$. Alternatively, by choosing a suitable
projection of $\operatorname{\Lambda}_{i}^{\wedge}$ onto a $d_{i}$-dimensional
subspace, one obtains a $k_{i}$-to-$d_{i}$ cut and project scheme with the
corresponding complexity and repetitivity function, up to linear constants.

##  4\. Acceptance domains and cut regions in the subsystems

In this section we shall see how properties of the subsystems relate to those
of the original cut and project scheme. We note that, in this section, it will
not be necessary to assume that the subsystems have constant stabiliser rank
(although we will still always assume that $\Gamma_{1}+\cdots+\Gamma_{m}$ is
finite index in $\Gamma$, as in Definition 3.25). Firstly, we observe that the
properties of hyperplane spanning and weak homogeneity are inherited:

######  Lemma 4.1.

A scheme $\mathcal{S}$ is hyperplane spanning if and only if each subsystem is
hyperplane spanning.

###### Proof.

Let $H_{X}=H\cap X_{H}\in\mathscr{H}_{i}$, where $H\in\mathscr{H}$ and
$X_{H}=X_{i}$ for some $i$. We let $V_{X}$ be the translate of $H_{X}$ over
the origin (see Notation 3.16). Suppose that $\mathcal{S}$ is hyperplane
spanning. Then $\Gamma_{<}^{H}=(\Gamma_{<})\cap V$ has linear span all of
$V\coloneqq V(H)$. By assumption,
$\Gamma^{\prime}\coloneqq(\Gamma_{1})_{<}+(\Gamma_{2})_{<}+\cdots+(\Gamma_{m})_{<}$
is finite index in $\Gamma$, so $\Gamma^{\prime}\cap\Gamma^{H}$ still spans
$V$. By Lemma 3.17, we may write $V=V_{X}+\widehat{X}_{i}$. Write
$\widehat{\Gamma}_{i}$ for the sum of all $\Gamma_{j}$, with $j\neq i$. By
definition, $(\widehat{\Gamma}_{i})_{<}\leqslant\widehat{X}_{i}$, which is
complementary with $V_{X}$. So $\Gamma^{\prime}\cap
V=(\Gamma_{i})_{<}+(\widehat{\Gamma}_{i})_{<}$ can only have linear span all
of $V$ if $(\Gamma_{i})_{<}$ has linear span all of $V_{X}$, since $V_{X}$ is
complementary with $\widehat{X}_{i}$ in $V$. Since $H$ was arbitrary, we see
that each subsystem is hyperplane spanning, as required.

Suppose, on the other hand, that each subsystem is hyperplane spanning. Then
for any $H_{X}\in\mathscr{H}_{i}$, the stabiliser $(\Gamma_{i}^{H_{i}})_{<}$
has linear span all of $V_{X}$. Since each $(\Gamma_{j})_{<}$ projects densely
into $X_{j}$, we have that $(\widehat{\Gamma}_{i})_{<}$ spans
$\widehat{X}_{i}$. It follows that
$\Gamma_{i}^{H_{i}}+\widehat{\Gamma}_{i}\leqslant\Gamma$ has projection
spanning all of $V=V_{X}+\widehat{X}_{i}$, so that $\mathcal{S}$ is hyperplane
spanning, as required. ∎

######  Lemma 4.2.

The scheme $\mathcal{S}$ is weakly homogeneous if and only if the subsystems
$(X_{i},\Gamma_{i},W_{i})$ are weakly homogeneous.

###### Proof.

First, assume that $\mathcal{S}$ is weakly homogeneous. We take $o$ as the
origin in $\operatorname{\mathbb{E}_{<}}$ (which we can do without loss of
generality, since homogeneity is independent of translation of $W$). Hence,
for each $H\in\mathscr{H}$, we have that
$V=V(H)=H+(1/n_{H})\cdot(\gamma_{H})_{<}$ for some $\gamma_{H}\in\Gamma$. By
assumption, we have that
$\Gamma\leqslant\frac{1}{N}\Gamma_{1}+\cdots+\frac{1}{N}\Gamma_{m}$ for some
$N\in\mathbb{N}$, so that
$\gamma_{H}=\frac{\gamma_{1}}{N}+\cdots+\frac{\gamma_{m}}{N}$ for
$\gamma_{i}\in\Gamma_{i}$. Given $X_{H}=X_{i}$, by Lemma 3.17,

$V_{X}+\widehat{X}_{i}=H_{X}+\widehat{X}_{i}+\left(\frac{\gamma_{1}}{N\cdot
n_{H}}\right)_{<}+\cdots+\left(\frac{\gamma_{m}}{N\cdot
n_{H}}\right)_{<}=H_{X}+\widehat{X}_{i}+\left(\frac{\gamma_{i}}{N\cdot
n_{H}}\right)_{<}.$

The final equality above follows from the fact that
$(\gamma_{j})_{<}\in\widehat{X}_{i}$, for $j\neq i$. It follows that
$H_{X}+(\gamma_{i}/(N\cdot n_{H}))_{<}$ contains the origin, since
$\widehat{X}_{i}$ is complementary to $X_{i}\supseteq H_{X}$. Since
$H\in\mathscr{H}$ was arbitrary, we see that all $H_{X}\in\mathscr{H}_{i}$ may
be translated over the origin, by some element $(\gamma_{i}/N\cdot n_{H})$, so
each subsystem is weakly homogeneous.

Conversely, suppose that each subsystem is weakly homogeneous and let
$H\in\mathscr{H}$, with $H_{X}\in\mathscr{H}_{i}$ for some $i$. Again, we
assume that the origin $o$ is taken as the origin for each $W_{i}$ (and we can
translate each $W_{i}$ independently, since whichever translates we choose
their sum will still be a translate of $W$). There exists some
$\gamma\in\Gamma_{i}$ and $n\in\mathbb{N}$ with $H_{X}+(\gamma/n)_{<}$
containing the origin. Then
$H+(\gamma/n)_{<}=H_{X}+\widehat{X}_{i}+(\gamma/n)_{<}$ contains the origin.
Since $H\in\mathscr{H}$ was arbitrary, we see that $\mathcal{S}$ is weakly
homogeneous. ∎

If one assumes that each subsystem $(X_{i},\Gamma_{i},W_{i})$ is strictly
homogeneous, then it is easy to see in the above proof that $\mathcal{S}$ is
homogeneous. Conversely, if $\mathcal{S}$ is homogeneous (so we may take each
$n_{H}=1$), then we use the lattice elements
$\gamma_{i}/N\in\frac{1}{N}\Gamma_{i}$ in the above proof. Hence, the
subsystems $(X_{i},\frac{1}{N}\Gamma_{i},W_{i})$ are homogeneous. There is
little utility in saying more since, even assuming C, we cannot assume in
general that $N=1$ i.e., that $\Gamma_{1}+\cdots+\Gamma_{m}=\Gamma$.

Recall from Section 3.2.1 that we still have notions of cut regions for
subsystems. We set their notation here:

######  Notation 4.3.

For each $i=1$, …, $m$ and $r>0$ we define the $r$-cut regions
$\mathscr{C}_{i}(r)$ of the systems $(X_{i},\frac{1}{N}\Gamma_{i},W_{i})$
analogously to the usual cut regions (note here that we use the lattices
$\frac{1}{N}\Gamma_{i}$, whose sum contains $\Gamma$). In more detail, the
$r$-cut regions $\mathscr{C}_{i}(r)$ are defined as the connected components
of $W_{i}$ after removing all translates $H+\frac{\gamma_{<}}{N}$, where
$H\in\mathscr{H}_{i}$ and $\gamma\in\Gamma_{i}(r)$ (recall that
$\Gamma_{i}(r)\coloneqq B_{r}(0)\cap\Gamma_{i}$).

If it were the case that $\Gamma_{1}+\cdots+\Gamma_{m}=\Gamma$ (rather than
just being a finite index subgroup) and $N=1$, then hyperplanes
$H\in\mathscr{H}$ would be aligned to all but one factor $X_{i}$ of the
decomposition. So it is easy to see that in that case the $r$-cut regions for
$W$ would correspond to products of $r$-cut regions in each $W_{i}$ for the
subsystems. In general, we can not assume that $N=1$; however, in the next
lemma we see that products of cut regions refine those of the whole window,
allowing us to work with the finer lattices $\frac{1}{N}\Gamma_{i}$ for the
cut regions.

######  Lemma 4.4.

There is some $\lambda>0$ satisfying the following. For each
$C\in\mathscr{C}(r)$, there are $C_{i}\in\mathscr{C}_{i}(\lambda r)$ with
$C_{1}+C_{2}+\cdots+C_{m}\subseteq C$.

###### Proof.

Since $\Gamma_{1}+\cdots+\Gamma_{m}$ is finite index in $\Gamma$, there exists
some $\nu>0$ so that whenever $\|\gamma_{1}+\cdots+\gamma_{m}\|\leq r$, for
$\gamma_{i}\in\Gamma_{i}$, then each $\|\gamma_{i}\|\leq\nu r$ (indeed, note
that the linear spans of the $\Gamma_{i}$ give a direct sum decomposition of
$\operatorname{\mathbb{E}}$).

Let $\mathscr{H}^{\pm}$ denote the set of open half spaces $(H^{+})^{\circ}$
and their opposites $(H^{+})^{c}$ for $H^{+}\in\mathscr{H}^{\pm}$, and
similarly for $\mathscr{H}_{i}^{\pm}$. We may alternatively express $C$ as an
intersection of half spaces

$C=\bigcap_{Z\in\mathscr{H}^{\pm}}Z+(\gamma_{Z})_{<},$

where each $\gamma_{Z}\in\Gamma(r)$. By Lemma 3.17, we may write each
$Z\in\mathscr{H}_{i}^{\pm}$ as $Z=Z_{X}+\widehat{X}_{i}$, where
$Z_{X}\in\mathscr{H}_{i}^{\pm}$. Moreover, we may write each
$\gamma_{Z}\in\Gamma(r)$ as

$\gamma_{Z}=\frac{\gamma_{1}}{N}+\frac{\gamma_{2}}{N}+\cdots+\frac{\gamma_{m}}{N},$

where $\gamma_{i}\in\Gamma_{i}$. In fact, each $\gamma_{i}\in\Gamma_{i}(N\nu
r)$, since $\|\gamma_{1}+\cdots+\gamma_{m}\|=\|N\cdot\gamma_{Z}\|\leq Nr$.
Hence, we may rewrite the above intersection defining $C$ as

$C=\bigcap_{i=1}^{m}\bigcap_{Z_{X}\in\mathscr{H}_{i}^{\pm}}(Z_{X}+\widehat{X}_{i})+(\gamma_{Z})_{<}=\bigcap_{i=1}^{m}\bigcap_{Z_{X}\in\mathscr{H}_{i}^{\pm}}Z_{X}+\widehat{X}_{i}+\left(\frac{\gamma_{i}}{N}\right)_{<},$

since each $(\gamma_{j})_{<}\in\widehat{X}_{i}$, for $j\neq i$. Since each
$X_{i}$ is complementary to $\widehat{X}_{i}$ this is equivalent to

$C=C_{1}+C_{2}+\cdots+C_{m},\ \text{ where }\
C_{i}=\bigcap_{Z_{X}\in\mathscr{H}_{i}^{\pm}}Z_{X}+\left(\frac{\gamma_{i}}{N}\right)_{<}.$

Each $C_{i}$ is an intersection of half spaces, or their opposites, from
$\mathscr{H}_{i}^{\pm}$, and each $\gamma_{i}\in\Gamma_{i}(N\nu r)$. It
follows that each $C_{i}$ is either a cut region of $\mathscr{C}_{i}(N\nu r)$,
or contains such a cut region. By taking such a cut region for each $C_{i}$,
and letting $\lambda=N\nu$, the result follows. ∎

The above shows that if each $r$-cut region in every subsystem is large, then
all cut regions in the whole system are large. We will show later that this
happens if we assume C and D for our subsystems. Low complexity is inherited
from the subsystems, using the generalised complexity result of Theorem 2.2:

######  Proposition 4.5.

For each $i=1$, …, $m$, let $\alpha_{i}$ be the complexity exponent for each
subsystem, calculated as stated in Theorem 2.2. Then the complexity exponent
of the whole system is given by $\alpha=\sum_{i=1}^{m}\alpha_{i}$. Moreover,
$\mathcal{S}$ satisfies C if and only if each subsystem satisfies
$\alpha_{i}=d_{i}$.

###### Proof.

Fix some $i=1$, …, $m$. We wish to compute $\operatorname{rk}(H_{X})$ and
$\beta_{H_{X}}$ for each $H_{X}\in\mathscr{H}_{i}$. Firstly, we have that
$\Gamma^{H_{X}}+\widehat{\Gamma}_{i}$ is a finite index subgroup of
$\Gamma^{H}$, where $\widehat{\Gamma}_{i}$ is the sum of $\Gamma_{j}$ with
$j\neq i$. To see this, first note that by assumption
$\Gamma_{i}+\widehat{\Gamma}_{i}$ is finite index in $\Gamma$, so that
$\operatorname{rk}(H)=\operatorname{rk}(\Gamma\cap
V)=\operatorname{rk}((\Gamma_{i}+\widehat{\Gamma}_{i}))_{<}\cap V)$. By Lemma
3.17, $V=V(H)=V_{X}+\widehat{X}_{i}$. Since $V_{X}\supseteq(\Gamma_{i})_{<}$
and $\widehat{X}_{i}\supseteq(\widehat{\Gamma}_{i})_{<}$ give a sum
decomposition of $V$, we have that

$(\Gamma_{i}+\widehat{\Gamma}_{i})_{<}\cap V=((\Gamma_{i})_{<}\cap
V_{X})+((\widehat{\Gamma}_{i})_{<}\cap\widehat{X}_{i})=(\Gamma_{i}^{H_{X}})_{<}+(\widehat{\Gamma}_{i})_{<}.$

The last sum is of two complementary subgroups, so we see that

(4.1)
$\operatorname{rk}(H)=\operatorname{rk}_{i}(H_{X})+\operatorname{rk}(\widehat{\Gamma}_{i})=\operatorname{rk}_{i}(H_{X})+(k-k_{i}),\
\text{ that is }\
\operatorname{rk}_{i}(H_{X})=\operatorname{rk}(H)-(k-k_{i}).$

To determine $\dim(\langle\Gamma^{H_{X}}\rangle_{\mathbb{R}})$, we note by the
same reasoning as above,

$\langle\Gamma^{H}_{<}\rangle_{\mathbb{R}}=\langle\Gamma^{H_{X}}_{<}\rangle_{\mathbb{R}}+\widehat{X}_{i},$

since $(\widehat{\Gamma}_{i})_{<}$ is dense in $\widehat{X}_{i}$. This is a
direct sum decomposition, so

(4.2) $\beta_{H}=\beta_{H_{X}}+(n-n_{i}),\ \text{ that is }\
\beta_{H_{X}}=\beta_{H}-(n-n_{i}).$

Using Equations 4.1 and 4.2, for a flag $f_{i}\subseteq\mathscr{H}_{i}$ we
calculate

(4.3) $\displaystyle\alpha_{f_{i}}=$ $\displaystyle\sum_{H_{X}\in
f_{i}}d_{i}-\operatorname{rk}(H_{X})+\beta_{H_{X}}$ (4.4)
$\displaystyle=\sum_{H_{X}\in
f_{i}}(k_{i}-n_{i})-(\operatorname{rk}(H)-(k-k_{i}))+(\beta_{H}-(n-n_{i}))$
$\displaystyle=\sum_{H_{X}\in f_{i}}d-\operatorname{rk}(H)+\beta_{H}.$

In other words, we get the same sum for the flag $f_{i}$ as one does by
considering it as a subset of a flag in the whole system, replacing each
$H_{X}\in\mathscr{H}_{i}$ with $H\in\mathscr{H}$. It is not hard to see that
choosing a flag $f\subseteq\mathscr{H}$ is equivalent to choosing flags
$f_{i}\subseteq\mathscr{H}_{i}$, one for each $i=1$, …, $m$. Indeed, after
repositioning the hyperplanes over the origin,

$\bigcap_{V\in f}V=\bigcap_{i=1}^{m}\bigcap_{V_{X}\in
f_{i}}(V_{X}+\widehat{X}_{i})=\bigcap_{i=1}^{m}Z_{i}+\widehat{X}_{i},\ \
\text{ where }Z_{i}=\bigcap_{V_{X}\in f_{i}}V_{X}.$

Each $Z_{i}$ is trivial if and only if each $f_{i}$ is a flag. In this case,
the above intersection is of the $\widehat{X}_{i}$, which is trivial, so $f$
is a flag. Conversely, if some $f_{i}$ is not a flag, then $Z_{i}$ is non-
trivial. Since $Z_{i}\subseteq\widehat{X}_{j}$ for all $j\neq i$, we see that
the intersection of hyperplanes is non-trivial, so $f$ is not a flag.

It follows that

$\alpha=\max_{f\in\mathscr{F}}\alpha_{f}=\max_{f\in\mathscr{F}}\left(\sum_{i=1}^{m}\alpha_{f_{i}}\right)=\sum_{i=1}^{m}\max_{f_{i}\in\mathscr{F}_{i}}\alpha_{f_{i}}=\sum_{i=1}^{m}\alpha_{i},$

where $\mathscr{F}$ is the set of flags of $\mathscr{H}$, $\mathscr{F}_{i}$ is
the set of flags of $\mathscr{H}_{i}$ and, for the second equality above,
given a flag $f$ we let $f_{i}$ be the associated flag in $\mathscr{H}_{i}$.
Since each subsystem can be regarded as an individual cut and project scheme,
Corollary 2.5 still applies and each $\alpha_{f_{i}}\geq d_{i}$. It follows
that $\alpha=d$ if and only if each $\alpha_{i}=d_{i}$, since
$d=d_{1}+\cdots+d_{m}$. ∎

For minimal complexity schemes, when the subsystems have constant stabiliser
ranks $r_{i}$ then we may determine these ranks from only the ranks of the
$\Gamma_{i}$ and dimensions of the $X_{i}$:

######  Proposition 4.6.

Suppose that $\mathcal{S}$ has property C and is equipped with a decomposition
with constant stabiliser ranks. Then each such rank is given by

$r_{i}=k_{i}-\delta_{i}-1,\ \text{ where
}\delta_{i}\coloneqq\frac{d_{i}}{n_{i}}.$

###### Proof.

By the above Proposition 4.5, each subsystem has property C, and is also
hyperplane spanning by Theorem 2.8. By Theorem 2.2, we see that

$\sum_{V\in f_{i}}(d_{i}-r_{i}+(n_{i}-1))=\sum_{V\in
f_{i}}(k_{i}-r_{i}-1)=d_{i},$

for each flag $f_{i}\subseteq\mathscr{H}_{i}$. Since each $r_{i}$ is constant,
and there are $n_{i}$ elements in a flag, we deduce

$n_{i}(k_{i}-r_{i}-1)=d_{i}\ \text{ that is, }\ r_{i}=k_{i}-\delta_{i}-1.$

∎

Notice that if $\mathcal{S}$ has property C and already has constant
stabiliser ranks, then by the above this rank is given by
$k-\delta-1\in\mathbb{N}$, and hence $n$ must divide $d$ (see also [FHK02,
Theorem 6.7]). This is the case, for example, when $\mathcal{S}$ is
indecomposable.

We saw in Lemma 4.4 that products of cut regions for the subsystems
$(X_{i},\frac{1}{N}\Gamma_{i},W_{i})$ refine the cut regions of the original
scheme. In particular, if each such subsystem has large cut regions, then so
does the whole system. We now consider the acceptance domains. For proving
that LR implies C and D, we will show that if some subsystem has a small
acceptance domain (in terms of its volume) then so does the whole scheme. For
this reason, for defining the acceptance domains of each subsystem, we use the
subsystems $(X_{i},\Gamma_{i},W_{i})$ (that is, using $\Gamma_{i}$ rather than
$\frac{1}{N}\Gamma_{i}$), whose products of acceptance domains contain those
of the whole system:

######  Notation 4.7.

We define the $r$-acceptance domains for a subsystem
$(X_{i},\Gamma_{i},W_{i})$ in the usual way. That is, (for sufficiently large
$r$) the $r$-acceptance domain $A\subseteq W_{i}$ containing $x\in W_{i}$ (for
$x\notin\partial W_{i}+(\Gamma_{i})_{<}$) is the intersection of
$(\Gamma_{i}(r))_{<}$ translates of $W_{i}^{\circ}$ and $W_{i}^{c}$ which
contain $x$ (see [KW20, Corollary 3.2]). The finite set of possible
$r$-acceptance domains is denoted $\mathscr{A}_{i}(r)$.

######  Lemma 4.8.

There is some $c>0$ so that, for sufficiently large $r$, given
$A_{i}\in\mathscr{A}_{i}(r)$ for each $i=1$, …, $m$, there exists
$A\in\mathscr{A}(r+c)$ for which $A\subseteq A_{1}+A_{2}+\cdots+A_{m}$.

###### Proof.

Write each $A_{i}$ as an intersection

$A_{i}=\left(\bigcap_{\gamma\in
P_{i}}(W_{i}^{\circ}+\gamma_{<})\right)\cap\left(\bigcap_{\gamma\in
Q_{i}}(W_{i}^{c}+\gamma_{<})\right)$

for subsets $P_{i},Q_{i}\subseteq\Gamma_{i}(r)$, where $A_{i}\subseteq
W_{i}^{\circ}$ (as $0\in P_{i}$). We now simply translate $W^{\circ}$ and
$W^{c}$ by the same lattice elements and take the corresponding intersection.
Since each $\gamma\in\Gamma_{i}\subseteq X_{i}$, it is not hard to see that

(4.5) $(A_{i}+\widehat{X}_{i})\cap W^{\circ}=\left(\bigcap_{\gamma\in
P_{i}}(W^{\circ}+\gamma_{<})\right)\cap\left(\bigcap_{\gamma\in
Q_{i}}(W^{c}+\gamma_{<})\right)\cap W^{\circ}.$

Indeed, an element of $W^{\circ}$ belongs to either intersection precisely
when its projection to $X_{i}$ is in $A_{i}$, since each $\gamma_{<}\in
X_{i}$.

We have that

$A_{1}+\cdots+A_{m}=\bigcap_{i=1}^{m}(A_{i}+\widehat{X}_{i})=\bigcap_{i=1}^{m}((A_{i}+\widehat{X}_{i})\cap
W^{\circ}),$

with the last equality coming from the fact that each $A_{i}\subseteq
W_{i}^{\circ}$, so that any element in $A_{1}+\cdots+A_{m}$ must be in
$W^{\circ}$. By Equation 4.5, the right-hand intersection is an intersection
of translates of $W^{\circ}$ or $W^{c}$ by elements $\gamma_{<}$ for
$\gamma\in\Gamma_{i}(r)\leqslant\Gamma(r)$. Each $A\in\mathscr{A}(r)$ is a
minimal intersection of such elements, possibly more, so there exists some
$A\subseteq A_{1}+\cdots+A_{m}$, as required. ∎

If we have that $\Gamma=\Gamma_{1}+\cdots+\Gamma_{m}$, then something may be
said about the the converse result of the above lemma, of finding a product of
acceptance domains $A_{1}+\cdots+A_{m}$ contained in some given
$A\in\mathscr{A}(r)$. However, the precise statement is slightly technical and
will not be needed, so we omit it here. The implication of the above lemma
that will be used in the proof of our main theorem is the following:

######  Corollary 4.9.

Suppose that for some subsystem $(X_{i},\Gamma_{i},W_{i})$, and any given
$\varepsilon>0$, we may find some $r>0$ and some acceptance domain
$A_{i}(\varepsilon)\in\mathscr{A}_{i}(r)$ with volume
$|A_{i}(\varepsilon)|<\varepsilon/r^{\alpha_{i}}$, where $\alpha_{i}$ is the
complexity exponent for this subsystem. Then, for every $\varepsilon>0$, we
may find an acceptance domain $A(\varepsilon)\in\mathscr{A}(r)$ with volume
$|A(\varepsilon)|<\varepsilon/r^{\alpha}$, where $\alpha$ is the complexity
exponent of $\mathcal{S}$.

###### Proof.

For each $j\neq i$, we may find some acceptance domain
$A_{j}\in\mathscr{A}(r)$ with volume $A_{j}\ll 1/r^{\alpha_{j}}$, since
$\\#\mathscr{A}_{j}(r)\asymp r^{\alpha_{j}}$, and the volumes of the
acceptance domains $\mathscr{A}_{j}(r)$ sum to the volume of $W_{j}$. Since
$\\{X_{i}\\}_{i=1}^{m}$ is a direct sum decomposition of
$\operatorname{\mathbb{E}_{<}}$, we may find some $C>0$ so that for any
$\varepsilon>0$ there are acceptance domains with

$|A_{1}+\cdots+A_{i}(\varepsilon)+\cdots+A_{m}|\leq
C\varepsilon\cdot\frac{1}{r^{\alpha_{1}}}\cdot\frac{1}{r^{\alpha_{2}}}\cdot\cdots\cdot\frac{1}{r^{\alpha_{m}}}=\frac{\varepsilon}{r^{\alpha}}.$

In the above, we use the fact that $\alpha=\sum\alpha_{i}$ by Proposition 4.5.
By Lemma 4.8, there exists some $A\in\mathscr{A}(r)$ with $A\subseteq
A_{1}+\cdots+A_{m}$ so that $|A|\leq C\varepsilon/r^{\alpha}$. Since $C$ does
not depend on $\varepsilon>0$, which can be taken arbitrarily small, the
result follows. ∎

######  Remark 4.10.

The notion of a system satisfying _positivity of weights_ (PW) will be
introduced in Section 6. With this terminology, the above theorem implies that
if one subsystem of a decomposition does not satisfy PW, then $\mathcal{S}$
does not satisfy PW.

##  5\. The Diophantine condition D

Now that we have defined what it means to decompose a cut and project scheme,
we can define our Diophantine condition D. This will say that the lattice
points do not project close to the origin, relative to their norm in the total
space, in any subsystem. Equivalently, lattice points which lie close to the
physical space $\operatorname{\mathbb{E}_{\vee}}$ are necessarily distant from
the origin. Before specialising to cut and project sets, we first establish
some basic properties for the abstract notion of a Diophantine lattice.

###  5.1. Diophantine lattices

Let $G$ be a free Abelian group of rank $k$. Call $\eta\colon G\to\mathbb{R}$
a lattice norm if there exists an embedding of $G$ as a lattice into a normed
vector space (of dimension $k$) so that $\eta$ is the restriction of this norm
to $G$. Since norms on finite dimensional vector spaces are equivalent, and
any two given lattices are linearly isomorphic, any two lattice norms
$\eta_{1}$ and $\eta_{2}$ are linearly equivalent, in the sense that there
exist constants $A$ and $B$ so that $\eta_{1}(g)\leq A\eta_{2}(g)$ and
$\eta_{2}(g)\leq B\eta_{1}(g)$ for all $g\in G$. It is not too hard to show
that a lattice norm is the same as a $\mathbb{Z}$-norm (satisfying the usual
properties of a norm, but with $\eta(\lambda\cdot g)=|\lambda|\cdot\eta(g)$
for $g\in G$ and only $\lambda\in\mathbb{Z}$) which gives $G$ the discrete
topology or, equivalently, is such that all $g\in G\setminus\\{0\\}$ have norm
$\eta(g)>c$ for some $c>0$. A standard choice of lattice norm is given by
choosing a $\mathbb{Z}$-basis $\\{g_{i}\\}_{i=1}^{k}$ for $G$, and defining
$\eta(\sum_{i=1}^{k}\lambda_{i}g_{i})\coloneqq\max|\lambda_{i}|$ for
$\lambda_{i}\in\mathbb{Z}$.

######  Definition 5.1.

Let $G$ be a free Abelian group of rank $k$, given as a dense subgroup
$G\leqslant X$ of some $n$-dimensional vector space $X$. Choose any norm on
$X$ and any lattice norm $\eta$ on $G$. Then we call $G$ Diophantine if there
exists some $c>0$ so that, for all $g\in G\setminus\\{0\\}$, we have

$\|g\|\geq c\cdot\eta(g)^{-\delta},\ \text{ where }\delta=\frac{k-n}{n}.$

We call $\delta=(k-n)/n=d/n$, for $d\coloneqq k-n$, the Diophantine exponent.
The Diophantine property depends only on the embedding $G\leqslant X$ up to
linear isomorphism, since all norms and all lattice norms are linearly
equivalent.

######  Example 5.2.

Let $\alpha\in\mathbb{R}$ be any irrational number. Consider the dense, rank
$2$, free Abelian group
$G(\alpha)\coloneqq\mathbb{Z}+\alpha\mathbb{Z}\leqslant\mathbb{R}$, where
$\mathbb{R}$ has the usual norm $\|x\|\coloneqq|x|$ and $G(\alpha)$ has
lattice norm $\eta(m+n\alpha)\coloneqq|m|+|n|$. Then $G(\alpha)$ is
Diophantine if for all $(m,n)\in\mathbb{Z}^{2}\setminus\\{(0,0)\\}$ we have

$|n\alpha-m|\geq\frac{c}{|m|+|n|}.$

We may as well only consider the above for $n\alpha-m$ small (relative to
$n$), so taking $m=[n\alpha]$, the nearest integer to $n\alpha$. Since
$[n\alpha]\asymp n$ we equivalently need some constant $\tau>0$ with

$d(n\alpha,\mathbb{Z})\geq\frac{\tau}{n},$

for all $n\in\mathbb{N}$, where $d(n\alpha,\mathbb{Z})$ denotes the distance
from $n\alpha$ to the nearest integer. Alternatively,

$\left|\alpha-\frac{m}{n}\right|\geq\frac{C}{n^{2}}$

for some constant $C>0$ and all $m\in\mathbb{Z}$, $n\in\mathbb{N}$. This is
the standard notion of a number $\alpha$ being badly approximable, which by a
classical result of Dirichlet is equivalent to $\alpha$ having continued
fraction expansion with bounded entries.

More generally, given vectors $\\{v_{i}\\}_{i=1}^{d}$, for
$v_{i}\in\mathbb{R}^{n}$, we may consider the group

$G(\alpha_{1},\ldots,\alpha_{d})\coloneqq\langle
v_{1},v_{2},\ldots,v_{d}\rangle_{\mathbb{Z}}+\mathbb{Z}^{n},$

assumed to be rank $k$ and dense in $\mathbb{R}^{n}$. This is Diophantine if
and only if the system of vectors is badly approximable in the usual sense
[Cas65].

In a previous work [HKW18] for ‘cubical’ cut and projects, there was a natural
choice for an integer reference lattice, so that Diophantine notions such as
badly approximable could be understood in terms of that frame of reference. In
our current more general setting of polytopal cut and project sets there is no
canonical candidate. Instead, given $G\leqslant X$, we may choose subgroups
$G^{\prime}$, $Z\leqslant G$ of ranks $k-n$ and $n$, respectively, so that the
‘reference lattice’ $Z$ spans $X$ and $G=G^{\prime}+Z$. There is a linear map
$A\colon X\to\mathbb{R}^{n}$, taking $Z$ to $\mathbb{Z}^{n}$, and $G^{\prime}$
to some rank $k-n$ subgroup $A(G^{\prime})=G(\alpha_{1},\ldots,\alpha_{d})$
for some $\alpha_{i}\in\mathbb{R}^{n}$. Since $(G\leqslant X)$ and
$(G(\alpha_{1},\ldots,\alpha_{d})+Z\leqslant\mathbb{R}^{n})$ are linearly
isomorphic, one is Diophantine if and only if the other is. We see that the
above example is essentially the general case. However, the choice of the
reference lattice $Z$ is inconsequential to the Diophantine property, so the
most elegant solution is to not choose one.

We note that the Diophantine property is stable under taking finite index
super- or subgroups:

######  Lemma 5.3.

Let $G\leqslant K$, where $G$ and $K$ are free Abelian groups of equal rank
$k$, densely embedded in some Euclidean space $X$. Then $G$ is Diophantine if
and only if $K$ is.

###### Proof.

Take any lattice norm $\eta$ on $K$. We may take the restriction of $\eta$ as
the lattice norm on $G$. Clearly if the larger subgroup $K$ is Diophantine,
then so is the smaller group $G$, taking the same Diophantine constant $c$.

For the converse, suppose that $G$ is Diophantine. Since $G$ and $K$ are equal
rank, $G$ is finite index in $K$, so there exists some $N\in\mathbb{N}$ with
$N\cdot K\leqslant G$. So, given any $g\in K$, we have that

$\|g\|=\frac{1}{N}\|N\cdot g\|\geq\frac{1}{N}c\cdot\eta(N\cdot
g)^{-\delta}=\left(\frac{c}{N^{1+\delta}}\right)\eta(g),$

so $K$ is Diophantine with constant $c/N^{1+\delta}$. ∎

We have the following simple result, similar to the classical Dirichlet
Approximation Theorem, although weaker in the sense that the constant depends
on $G$ (a more closely related statement could involve choosing a reference
lattice as outlined above serving as the integer lattice
$\mathbb{Z}^{n}\leqslant\mathbb{R}^{n}$, but we prefer to give the following,
simpler statement). It shows that the Diophantine exponent $\delta$ is
optimal:

######  Lemma 5.4.

For $G\leqslant X$ and $\delta$ as in Definition 5.1, there is a constant
$c>0$ so that

$\|g\|\leq c\cdot\eta(g)^{-\delta}\text{ for infinitely many }g\in G.$

###### Proof.

Choose any two complementary subgroups $G^{\prime}$, $Z\leqslant G$ such that
$G^{\prime}+Z=G$ and $\operatorname{rk}(Z)=n$, with $Z$ spanning $X$. Choose
any basis $\\{z_{i}\\}_{i=1}^{n}$ for $Z$, with associated fundamental domain
$F$. Given any $g\in G^{\prime}$, there is some $z_{g}\in Z$ with $g-z_{g}\in
F$. We have that $\eta(z_{g})\ll\|g\|\ll\eta(g)$, since $Z$ has a basis which
is also a basis for $X$. Since there are $\gg r^{d}$ (where $d=k-n$) elements
$g\in G^{\prime}$ with $\eta(g)\leq r$ it follows that there are $\gg r^{d}$
elements $g-z_{g}\in G\cap F$ with $\eta(g-z_{g})\ll r$. So we may find some
$\kappa>0$ for which there are at least $\kappa r^{d}$ elements $g\in G\cap F$
with $\eta(g)\leq\frac{r}{2}$ for sufficiently large $r$.

Since $F$ is a fixed, $n$-dimensional fundamental domain, we may cover it by
strictly less than $\kappa r^{d}$ balls of radius $(\frac{c}{2})\cdot
r^{-\delta}=(\frac{c}{2})\cdot r^{-d/n}$ for some $c>0$. By the Pigeonhole
Principle, there is some ball which contains two distinct elements $g$, $h\in
G\cap F$ with $\eta(g)$, $\eta(h)\leq\frac{r}{2}$. So $g-h\in G$, $\|g-h\|\leq
cr^{-\delta}$ and $\eta(g-h)\leq r$. Since $r$ can be made arbitrarily large,
the result follows. ∎

Sums of Diophantine lattices are still Diophantine when they have the same
exponent:

######  Lemma 5.5.

Take two finite rank, free Abelian dense subgroups $G_{1}\leqslant X_{1}$ and
$G_{2}\leqslant X_{2}$ of vector spaces $X_{1}$ and $X_{2}$ with Diophantine
exponents $\delta_{1}$ and $\delta_{2}$, respectively. If
$\delta_{1}\neq\delta_{2}$, then $G_{1}+G_{2}\leqslant X_{1}+X_{2}$ is not
Diophantine. If $\delta_{1}=\delta_{2}$, then $G_{1}+G_{2}\leqslant
X_{1}+X_{2}$ is Diophantine if and only if both $G_{1}\leqslant X_{1}$ and
$G_{2}\leqslant X_{2}$ are Diophantine.

###### Proof.

Suppose that $\delta_{1}\neq\delta_{2}$; without loss of generality,
$\delta_{1}>\delta_{2}$. Denote $\operatorname{rk}(G_{1})=k_{1}$,
$\operatorname{rk}(G_{2})=k_{2}$, $\dim(X_{1})=n_{1}$, $\dim(X_{2})=n_{2}$ and
$d_{i}=k_{i}-n_{i}$, so that $\delta_{i}=\frac{d_{i}}{n_{i}}$. The required
Diophantine exponent $\delta$ for $G_{1}+G_{2}$ is
$\frac{d_{1}+d_{2}}{n_{1}+n_{2}}<\delta_{1}$. By the above Lemma 5.4, there
exist infinitely many elements $g\in G_{1}\setminus\\{0\\}$ with $\|g\|\leq
c\eta(g)^{-\delta_{1}}$ for some $c>0$. Since $cr^{-\delta_{1}}<\varepsilon
r^{-\delta}$ for arbitrarily small $\varepsilon$, for sufficiently large $r$,
these elements also rule out $G_{1}+G_{2}$ from being Diophantine.

Suppose then that $\delta_{1}=\delta_{2}$. Hence the Diophantine exponent
$\delta$ for $G_{1}+G_{2}\leqslant X_{1}+X_{2}$ is also
$\delta_{1}=\delta_{2}=\frac{d_{1}+d_{2}}{n_{1}+n_{2}}$. Suppose that one of
$G_{1}$ or $G_{2}$ is not Diophantine; without loss of generality, $G_{1}$ is
not Diophantine. Hence, for all $\varepsilon>0$, there exists some $g\in
G_{1}\setminus\\{0\\}$ with $\|g\|<\varepsilon\eta(g)^{-\delta}$. The
corresponding elements of $G_{1}+G_{2}$ show that this group is also not
Diophantine. Conversely, suppose that both $G_{1}$ and $G_{2}$ are
Diophantine. Since $X_{1}$ and $X_{2}$ are complementary in the sum, we have
that $\|g_{1}+g_{2}\|\geq\kappa_{1}\cdot\max\\{\|g_{1}\|,\|g_{2}\|\\}$ and
$\max\\{\eta(g_{1}),\eta(g_{2})\\}\leq\kappa_{2}\cdot\eta(g_{1}+g_{2})$ for
some $\kappa_{1}$, $\kappa_{2}>0$, for all $g_{1}\in G_{1}$, $g_{2}\in G_{2}$.

Take any non-trivial $g_{1}+g_{2}\in G_{1}+G_{2}$. If $g_{2}=0$, then using
that $G_{1}$ is Diophantine:

$\|g_{1}+g_{2}\|=\|g_{1}\|\geq
c_{1}\eta(g_{1})^{-\delta}=c_{1}\cdot\eta(g_{1}+g_{2})^{-\delta},$

since $\delta_{1}=\delta_{2}=\delta$. Similarly, if $g_{1}=0$ then
$\|g_{1}+g_{2}\|\geq c_{2}\cdot\eta(g_{1}+g_{2})^{-\delta}$ for some
$c_{2}>0$. If $g_{1}$ and $g_{2}\neq 0$ then

$\displaystyle\|g_{1}+g_{2}\|\geq\kappa_{1}\max\\{\|g_{1}\|,\|g_{2}\|\\}\geq\kappa_{1}\max\\{c_{1}\eta(g_{1})^{-\delta_{1}},c_{2}\eta(g_{2})^{-\delta_{2}}\\}\geq$
$\displaystyle\kappa_{1}\min\\{c_{1},c_{2}\\}\min\\{\eta(g_{1})^{-\delta},\eta(g_{2})^{-\delta}\\}=\kappa_{1}c\cdot\max\\{\eta(g_{1}),\eta(g_{2})\\}^{-\delta}\geq(\kappa_{1}\kappa_{2}c)\cdot\eta(g_{1}+g_{2})^{-\delta},$

where $c=\min\\{c_{1},c_{2}\\}$ and we use $\delta_{1}=\delta_{2}=\delta>0$.
So $G_{1}+G_{2}$ is Diophantine with constant $\kappa_{1}\kappa_{2}c$, as
required. ∎

###  5.2. Transference

The Diophantine property ensures that elements with small lattice norm are
reasonably distant from the origin. This implies a dual property: all group
elements with norm less than $r$ do not leave large gaps between them on a
bounded subset, relative to $r$. In the classical setting, of families of
vectors in $\mathbb{R}^{n}$ relative to the integer lattice $\mathbb{Z}^{n}$,
this is sometimes known as _transference_ ; see, for example [Cas65] (although
we note that the term more usually refers to Diophantine properties of a
system of vectors and the transpose system). Given our alternative
formulation, in terms of Diophantine lattices, we will restate this property
and recall the standard proof of it, using Minkowski’s Second Theorem.
Throughout, we assume that $G\leqslant X$ and $\eta$ are as in Definition 5.1.

######  Definition 5.6.

Call $G\leqslant X$ densely distributed if there is some $c>0$ so that, for
all $r>0$, every point in the unit ball of $X$ is within distance $c\cdot
r^{-\delta}$ of some $g\in G$ with $\eta(g)\leq r$.

In the above, $\delta$ is the Diophantine exponent $\delta=\frac{k-n}{n}$, as
in Definition 5.1. As usual, the choice of norm on $X$ does not make a
difference, up to a change of the constant $c$. The choice of the unit ball
above is also arbitrary. Given any other bounded region $B$, since $G$ is
dense in $X$, we can use finitely many $G$-translates $g_{i}\in G$ applied to
the unit ball to cover $B$. So all points of $B$ are within distance $c\cdot
r^{-\delta}$ of some element $g+g_{i}$ with $\eta(g)\leq r$, hence
$\eta(g+g_{i})\leq\eta(g)+\eta(g_{i})\leq r+\tau$ for some constant $\tau$.
After reparametrising, we thus have:

######  Lemma 5.7.

If $G\leqslant X$ is densely distributed, then there is some $c>0$ satisfying
the following: for any given bounded region $B$ we have that every $b\in B$ is
within distance $c\cdot r^{-\delta}$ of some $g\in G$, with $\eta(g)\leq r$.

There are of order $r^{k-n}$ elements of $G$ with lattice norm bounded above
by $r$ in the unit ball of $X$. On the other hand, one may also find of order
$r^{k-n}$ disjoint balls of radius $r^{-\delta}$ contained in the unit ball.
So being densely distributed is a special property, implying that $G$ fills
$X$ efficiently with respect to the lattice norm. The following transference
theorem says that this property follows from the Diophantine property:

######  Theorem 5.8.

If $G\leqslant X$ is Diophantine then it is densely distributed.

We prove the above through two lemmas, where we temporarily make use of the
viewpoint of $G\leqslant X$ being the projection of a lattice to a lower
dimensional subspace. This may always be done, as follows. Choose some basis
$\\{g_{i}\\}_{i=1}^{k}$ of $G$ and consider the projection of $\mathbb{R}^{k}$
to $X$ defined by $e_{i}\mapsto g_{i}$, where $e_{i}$ is the $i$th standard
basis vector. This maps the integer lattice $\mathbb{Z}^{k}$ isomorphically to
$G$. We will use analogous notation for this projection as the projection to
internal space for cut and project schemes.

######  Lemma 5.9.

Let $L$ be a lattice in a normed vector space $E$. Suppose that $\|g\|\geq
s>0$ for all $g\in L\setminus\\{0\\}$. Then for all $z\in E$ there is some
$g\in L$ such that

$\|z-g\|\leq c\cdot s^{1-\dim E}\operatorname{covol}(L)$

where the constant $c$ only depends on $\dim E$.

###### Proof.

The $i$th successive minimum in Minkowski’s Second Theorem is defined as

$\lambda_{i}=\inf\\{\lambda>0\mid\text{there exist linearly independent
}(g_{1},\ldots,g_{i})\text{ with each }\|g_{i}\|\leq\lambda\\}.$

Then clearly $s\leq\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{k}$, where
$k=\dim E$. Minkowski’s Second Theorem provides constants $0<A<B$, depending
only on $\dim E$, such that

$A\operatorname{covol}(L)\leq\lambda_{1}\lambda_{2}\cdots\lambda_{k}\leq
B\operatorname{covol}(L).$

It follows that

$\lambda_{k}\leq\frac{B\operatorname{covol}(L)}{\lambda_{1}\lambda_{2}\cdots\lambda_{k-1}}\leq
B\cdot s^{1-k}\operatorname{covol}(L).$

In other words, there is a sublattice of $L$ with basis
$\\{b_{i}\\}_{i=1}^{k}$ with each $\|b_{i}\|\leq B\cdot
s^{1-k}\operatorname{covol}(L)$. This basis defines a fundamental domain $F$
whose translates by $\mathbb{Z}$-sums of the $b_{i}$ cover $E$. Moreover, $F$
has diameter bounded above by $\|b_{1}\|+\cdots+\|b_{k}\|\leq k\cdot B\cdot
s^{1-k}\operatorname{covol}(L)$. Hence, every element of $E$ is within this
distance of a lattice element, as required. ∎

The above result shows that, for lattices of fixed covolume, if there are no
short lattice elements then the lattice is reasonably dense. The following
standard argument then shows that if the projection of the lattice to some
‘internal space’ $X$ keeps points distant from the origin, then those
projecting to some reasonably large ball in a complementary ‘physical space’
project densely to the ‘internal space’:

######  Lemma 5.10.

Fix a normed vector space $E$ with complementary subspaces $P$, $X\leqslant
E$. Given $z=(p+x)\in E$, write $z_{\vee}=p$ and $z_{<}=x$. We denote $k=\dim
E$, $n=\dim X$ and $d=k-n=\dim P$. Finally, fix a lattice $L\leqslant E$. Then
there are constants $A$, $B$ satisfying the following: Suppose that
$\|g_{<}\|\geq s$ for all $g\in L\setminus\\{0\\}$, with $\|g\|\leq r$. Then
for all $p\in P$ and $x\in X$, there is some $g\in L$ with

  * •

$\|p-g_{\vee}\|\leq A\cdot r^{-(d-1)}\cdot s^{-n}$, and

  * •

$\|x-g_{<}\|\leq B\cdot r^{-d}\cdot s^{-(n-1)}$.

###### Proof.

Define a linear map $M\colon E\to E$ by setting
$M(p+x)\coloneqq\frac{p}{r}+\frac{2x}{s}$ for $p\in P$ and $x\in X$. Since all
norms are linearly equivalent, we may as well choose norm on $E$ coming from
an inner product making $P$ and $X$ orthogonal, so that

$\det M=r^{-d}\cdot 2^{n}\cdot s^{-n},\ \text{ hence }\
\operatorname{covol}M(L)=(r^{-d}\cdot 2^{n}\cdot
s^{-n})\operatorname{covol}L.$

Let $g\in L\setminus\\{0\\}$ with $\|g\|\leq r$. Then

$\|M(g)\|=\|M(g)_{\vee}+M(g)_{<}\|=\left\|\frac{g_{\vee}}{r}+\frac{2g_{<}}{s}\right\|\geq\frac{2\|g_{<}\|}{s}-\frac{\|g_{\vee}\|}{r}\geq
1,$

since by assumption $\|g_{<}\|\geq s$ and $\|g_{\vee}\|\leq\|g\|\leq r$. Take
any $p\in P$ and $x\in X$ and consider the point
$M(p+x)=\frac{p}{r}+\frac{2x}{s}$. By the above, may apply Lemma 5.9 to the
lattice $M(L)$ with $s=1$, so there is a constant $c>0$ (depending only on
$k$) and some $g\in L$ for which

$\left\|\left(\frac{p}{r}+\frac{2x}{s}\right)-M(g)\right\|\leq
c\cdot\operatorname{covol}M(L).$

We have that $M(g)=\frac{g_{\vee}}{r}+\frac{2g_{<}}{s}$, so by orthogonality
of $P$ and $X$:

$\left\|\left(\frac{p}{r}+\frac{2x}{s}\right)-M(g)\right\|=\left\|\frac{p}{r}-\frac{g_{\vee}}{r}\right\|+\left\|\frac{2x}{s}-\frac{2g_{<}}{s}\right\|=\frac{1}{r}\|p-g_{<}\|+\frac{2}{s}\|x-g_{<}\|.$

Since each summand is bounded above by $c\cdot\operatorname{covol}M(L)$, we
obtain:

$\frac{1}{r}\|p-g_{<}\|,\ \frac{2}{s}\|x-g_{<}\|\leq c\cdot(r^{-d}\cdot
2^{n}\cdot s^{-n})\operatorname{covol}L$

The result follows by multiplying the first term by $r$, and the second by
$\frac{s}{2}$. ∎

Applying the above to the case where the projected lattice is Diophantine
proves the transference result:

###### Proof of Theorem 5.8.

Let $G\leqslant X$ be Diophantine. As remarked following the statement of
Theorem 5.8, up to linear isomorphism $G\leqslant X$ may be given as the
projection of a lattice $L\leqslant E$, for a $k$-dimensional vector space
$E$, onto an $n$-dimensional vector space $X$. We may regard $X$ as a subspace
of $E$ (up to linear isomorphism), by setting $P$ as the kernel of the
projection and identifying $X$ with any subspace complementary to $P$. Choose
such an extension to a ‘total space’ and any norm on $E$, whose restriction to
$L$ gives a lattice norm on $G=L_{<}$, setting $\eta(g_{<})\coloneqq\|g\|$ for
$g\in L$.

Choose any $b$ in the unit ball of $X$ and $r>0$. Since $G\leqslant X$ is
Diophantine, there is some constant $c>0$ so that $\|g_{<}\|\geq
c\cdot\eta(g)^{-\delta}=c\cdot\|g\|^{-\delta}$ for all $g\in
L\setminus\\{0\\}$, where $\delta=\frac{d}{n}=\frac{k-n}{n}$. Let $s\coloneqq
c\cdot r^{-\delta}$ and apply Lemma 5.10 on $p\coloneqq 0$, $x\coloneqq b$ so
that

  * •

$\|g_{\vee}\|\leq A\cdot r^{-(d-1)}\cdot(c\cdot
r^{-\delta})^{-n}=(Ac^{-n})\cdot r$;

  * •

$\|b-g_{<}\|\leq B\cdot r^{-d}\cdot(c\cdot
r^{-\delta})^{-(n-1)}=(Bc^{-(n-1)})\cdot r^{-\delta}$,

for some $g\in L$. Since $b$ belongs to the unit ball, we have that