$\widehat{{\mathcal{U}}}_{1}\smallsetminus{\mathcal{D}}_{0}$ splits into three
bigons, corresponding with the intersections with
$\widehat{{\mathcal{V}}}_{1}$, $\widehat{{\mathcal{W}}}_{4}$ and
$\widehat{{\mathcal{V}}}_{3}$, so that the first two bigons only have one side
in $\widehat{{\mathcal{U}}}_{1}\smallsetminus{\mathcal{D}}_{0}$ and the third
one have the two sides. Figure 16 also tells us that the boundary of
$\widehat{{\mathcal{W}}}_{1}\smallsetminus{\mathcal{D}}_{0}$ splits into two
bigons, corresponding with the intersections with
$\widehat{{\mathcal{U}}}_{2}$, $\widehat{{\mathcal{V}}}_{4}$ and both bigons
only have one side in
$\widehat{{\mathcal{W}}}_{1}\smallsetminus{\mathcal{D}}_{0}$.

###### Proof.

The proof is analogous to the proof of Lemma 5.10, in which we used the
$I$-bundle structure to determine the bigonization of the boundary. It is
enough to prove the result for the sets
$\widehat{{\mathcal{U}}}_{1}\smallsetminus{\mathcal{D}}_{0}$ and
$\widehat{{\mathcal{W}}}_{1}\smallsetminus{\mathcal{D}}_{0}$ because the
action of the dihedral group implies the result for the remaining balls. In
this case, we use the $I$-bundle structure provided by Lemma 7.10. The
components of the boundary for
$\widehat{{\mathcal{U}}}_{1}\smallsetminus{\mathcal{D}}_{0}$ are:

  1. 1.

$\big{(}\widehat{{\mathcal{U}}}_{1}\cap\widehat{{\mathcal{V}}}_{1}\big{)}\smallsetminus{\mathcal{D}}_{0}$
obtained as the $I$-bundle restricted to the segment $z_{4}\in[-1,1]$, cut
with the condition $z_{4}\geq z_{3}$, consisting of convex $4$-segments. There
is a side removed in this bigon, corresponding with the condition
$z_{3}=z_{4}$.

  2. 2.

$\big{(}\widehat{{\mathcal{U}}}_{1}\cap\widehat{{\mathcal{W}}}_{4}\big{)}\smallsetminus{\mathcal{D}}_{0}$
obtained as the $I$-bundle restricted to the arc $|z_{4}|=1$, with an side
removed, corresponding with the condition $z_{1}=z_{3}$.

  3. 3.

$\big{(}\widehat{{\mathcal{U}}}_{1}\cap\widehat{{\mathcal{V}}}_{3}\big{)}\smallsetminus{\mathcal{D}}_{0}$
obtained as the top face of the $I$-bundle structure, that is when
$|z_{4}-z_{3}|=1$. This bigon has the two vertices removed, corresponding to
the conditions $z_{4}=1$ and $z_{4}=-1$.

The components of the boundary for
$\widehat{{\mathcal{W}}}_{1}\smallsetminus{\mathcal{D}}_{0}$ are:

  1. 1.

$\big{(}\widehat{{\mathcal{W}}}_{1}\cap\widehat{{\mathcal{U}}}_{2}\big{)}\smallsetminus{\mathcal{D}}_{0}$
obtained as the top face of the $I$-bundle structure, that is when
$|z_{3}-z_{2}|=1$. This bigon has one side removed, corresponding to the
condition $z_{4}\in[0,1]$, consisting of non-convex $4$-segments.

  2. 2.

$\big{(}\widehat{{\mathcal{W}}}_{1}\cap\widehat{{\mathcal{V}}}_{4}\big{)}\smallsetminus{\mathcal{D}}_{0}$
obtained as the $I$-bundle restricted to the arc $|z_{4}|=1$, with a side
removed, corresponding with the condition $z_{3}=z_{1}$.

∎

Using this combinatorics we can conclude the following

######  Corolary 7.13.

For every $0\leq s<1$, the set $h^{-1}(s)\smallsetminus{\mathcal{D}}_{s}$ is
homeomorphic to $\mathbb{R}^{3}\sqcup\mathbb{R}^{3}$.

###### Proof.

As observed in the proof of Proposition 7.9, for every $0<s<1$, we recall the
decomposition of $h^{-1}(s)$ into the sets
$\\{\textrm{cl}({\mathcal{U}}_{j}),\textrm{cl}({\mathcal{V}}_{j}),\textrm{cl}({\mathcal{W}}_{j})\\}$
given in Lemma 5.10 and combinatorically equivalent to the one given in
Theorem 5.12 and represented in Figure 10. From that description, we can see
that the set $\mathcal{D}_{s}$ is the suspension over a closed curve in the
regular cube shown in Figure 14, and so, ${\mathcal{D}}_{s}$ is homeomorphic
to $\mathbb{S}^{2}$. Moreover, such curve decomposes the cube into two
topological discs, so that $h^{-1}(s)\smallsetminus{\mathcal{D}}_{s}$ is a
suspension over two disjoint discs, which is homeomorphic to
$\mathbb{R}^{3}\sqcup\mathbb{R}^{3}$. The case $s=0$ follows from the fact
that the description of the decomposition of
$h^{-1}(0)\smallsetminus{\mathcal{D}}_{0}$ into the sets
$\\{\widehat{{\mathcal{U}}_{j}}\smallsetminus{\mathcal{D}}_{0},\widehat{{\mathcal{V}}_{j}}\smallsetminus{\mathcal{D}}_{0},\widehat{{\mathcal{W}}_{j}}\smallsetminus{\mathcal{D}}_{0}\\}$
given by Lemma 7.11 is combinatorically equivalent to the one given for every
other level set $h^{-1}(s)$. ∎

In particular, the boundary $\partial S(4)$ can be seen as $\partial
S(4)\smallsetminus{\mathcal{D}}_{0}\cong\mathbb{R}^{3}\sqcup\mathbb{R}^{3}$,
compactified with a wedge of spheres and a disc
${\mathcal{D}}_{0}\cong(\mathbb{S}^{2}\vee\mathbb{S}^{2})\cup\mathbb{D}^{2}$,
and therefore, $\partial S(4)$ is not homeomorphic to the sphere
$\mathbb{S}^{3}$.

###  7.3 Good asymptotic behaviour at the boundary

Corollary 7.13 suggests that we can extend the cone structure $S(4)\cong
C(\mathbb{S}^{3})$ to the boundary outside ${\mathcal{D}}$. In this section we
prove this, and the most difficult part is to give to the sets
$\\{\overline{{\mathcal{U}}_{j}}\smallsetminus{\mathcal{D}},\overline{{\mathcal{V}}_{j}}\smallsetminus{\mathcal{D}},\overline{{\mathcal{W}}_{j}}\smallsetminus{\mathcal{D}}\\}$,
the structure of a cylinder. We start by looking at an example of another type
of bad convergence and explain why this complicates the argument.

######  Example 7.14.

For every $4$-segment $Z=[0,1,x,y]\in\widehat{{\mathcal{U}}}_{1}$ with
$0<x,y<1$ and for every $0\leq r\leq y$, there exist a sequence of simple
quadrilaterals $Z_{k}=[0,1,z_{k},w_{k}]\in\textrm{cl}({\mathcal{U}}_{1})$ such
that $h(Z_{k})=s_{k}$, $Z_{k}\rightarrow Z$ as $k\rightarrow\infty$, and the
fibers of the $I$-bundle in $\textrm{cl}({\mathcal{U}}_{1})\cap h^{-1}(s_{k})$
at $Z_{k}$, denoted as $\gamma_{z_{4}(k)}(t)$, satisfy that
$\gamma_{z_{4}(k)}(t)$ converge to the interval $[r,y+1]$. Thus, different
sequences converging to $Z$, lie on fibers which converge to different curves
at the boundary. The sequence can be constructed as follows: for every
$0<s<1$, the complex number $y+is=\eta e^{i\theta}$ has an element in the
bisector of the angle $\theta$ satisfying $\nu e^{i\theta/2}=r+is^{\prime}$
for some $s^{\prime}\in(0,s)$, then the sequence can be constructed with the
quadrilaterals
$Z_{s^{\prime}}=[0,1,x+is^{\prime},y+is]\in\textrm{cl}({\mathcal{U}}_{1})\cap
h^{-1}(s^{\prime})$, as $Z_{s^{\prime}}\rightarrow Z$ when $s\rightarrow 0$
and the fiber of the $I$-bundle structure at the point $z_{4}=y+is$ starts
with the segment $\\{t+is^{\prime}:r\leq t\leq y\\}$.

######  Remark 7.15.

Recall from the proof of Lemma 6.1, where we constructed homeomorphisms
$f_{s}:\textrm{cl}({\mathcal{U}}_{1})\cap h^{-1}(s)\rightarrow\mathbb{D}^{3}$,
induced by the $I$-bundle structure, which vary continuously on the parameter
$s\in(0,1)$ and gives us the cylinder structure

$\textrm{cl}({\mathcal{U}}_{1})\smallsetminus\\{\mathfrak{R}_{4}\\}\rightarrow\mathbb{D}^{3}\times(0,1),\quad
Z\mapsto(f_{h(Z)}(Z),h(Z)).$

If we restrict these homeomorphisms, we get homeomorphisms
$f_{s}:(\textrm{cl}({\mathcal{U}}_{1})\smallsetminus{\mathcal{D}})\cap
h^{-1}(s)\rightarrow\mathbb{H}^{2}\times(0,1]\cong\mathbb{H}^{3}$, then by
Lemma 5.10 and Lemma 7.11, these homeomorphisms are combinatorically
equivalent to the homeomorphism at the boundary
$f_{0}:\widehat{{\mathcal{U}}}_{1}\smallsetminus{\mathcal{D}}_{0}\rightarrow\mathbb{H}^{2}\times(0,1]$.
For the particular case of $4$-segments we can write
$f_{0}([0,1,x,y])=\left(y^{\prime},\frac{x}{y+1}\right)$ (for some
$y^{\prime}\in\mathbb{H}^{2}$) and Example 7.14 tells us that for every
$r\in[0,y]$, there is a sequence $\\{Z_{k}\\}$, such that
$Z_{k}\rightarrow[0,1,x,y]$ and
$\lim_{k\rightarrow\infty}f_{s_{k}}(Z_{k})=\left(y^{\prime},\frac{x-r}{y+1-r}\right)$,
so the extended family of functions $\\{f_{s}\\}_{s\in[0,1)}$ is no longer
continuous when $s\rightarrow 0$. In fact, there is no reasonable way to
define $f_{0}$ using the $I$-bundle structure to obtain such continuity. The
problem is of course the lack of uniqueness of convergence of fibers of the
$I$-bundle structures when converging to $4$-segments. But, if we restrict the
$I$-bundle structure on each level set
$h^{-1}(s)\cap\overline{{\mathcal{U}}_{1}}$, so that we have uniqueness on the
convergence of fibers, then we have a cylinder structure extended to the
boundary. Analogous behaviour occur for the $I$-bundle structures on the rest
of the sets
$\\{h^{-1}(s)\cap\overline{{\mathcal{U}}_{j}},h^{-1}(s)\cap\overline{{\mathcal{V}}_{j}},h^{-1}(s)\cap\overline{{\mathcal{W}}_{j}}:1\leq
j\leq 4\\}$.

We overcome the difficulty illustrated by Remark 7.15 and Example 7.14 by
proving regularity on different regions and gluing the pieces with Theorem A.1

######  Lemma 7.16.

The sets $\overline{{\mathcal{U}}_{j}}\smallsetminus{\mathcal{D}}$,
$\overline{{\mathcal{V}}_{j}}\smallsetminus{\mathcal{D}}$, and
$\overline{{\mathcal{W}}_{j}}\smallsetminus{\mathcal{D}}$, are homeomorphic to
$\mathbb{H}^{4}$.

###### Proof.

As it was done in multiple times already, thanks to the action of the dihedral
group $\langle\sigma,\tau\rangle$, it is enough to prove the result for
$\overline{{\mathcal{U}}_{1}}\smallsetminus{\mathcal{D}}$ and
$\overline{{\mathcal{W}}_{1}}\smallsetminus{\mathcal{D}}$.

Unlike previous proofs, we start here with the set
$\overline{{\mathcal{W}}_{1}}\smallsetminus{\mathcal{D}}$, because in this
case, it is easier. Recall from the construction on Lemma 6.1, that we can
uniformize the $I$-bundle structure of the cone
$\textrm{cl}({\mathcal{W}}_{1})$ on each level set $h^{-1}(s)$, and construct
the function

$\Omega\times[0,1]\rightarrow\textrm{cl}({\mathcal{W}}_{1})\cap
h^{-1}(s),\qquad(w,t)\mapsto[0,1,\gamma_{\varphi_{s}(w)}(t),\varphi_{s}(w)]$

(the curves $\gamma_{z}$ are the fibers of the $I$-bundle on
$\textrm{cl}({\mathcal{W}}_{1})\cap h^{-1}(s)$ and $\varphi_{s}$ is an
homeomorphism between $\Omega=\\{z\in\mathbb{D}^{2}:\textrm{Im}(z)\geq 0\\}$
and the base space of the bundle, cf. Figure 7). Taking into account the
collapsed fibers at the boundary, the previous function induces the
homeomorphism

$f_{s}:\textrm{cl}({\mathcal{W}}_{1})\cap
h^{-1}(s)\rightarrow\frac{\left(\Omega\times[0,1]\right)}{\sim}\cong\mathbb{D}^{3},$

which varies continuously on the parameter $s\in(0,1)$. We can restrict this
homeomorphism to $\Omega^{\prime}=\\{z\in\Omega:\textrm{Re}(z)\leq 0,\
\textrm{Im}(z)>0\\}\cong\mathbb{H}^{2}$, and observe that

$\big{(}\Omega^{\prime}\times(0,1]\big{)}\times(0,1)\rightarrow\overline{{\mathcal{W}}_{1}}\smallsetminus{\mathcal{D}},\qquad(w,t,s)\mapsto[0,1,\gamma_{\varphi_{s}(w)}(t),\varphi_{s}(w)],$

defines an homeomorphism onto its image, giving us the homeomorphism
$X=\\{[0,1,z,w]\in\overline{{\mathcal{W}}_{1}}\smallsetminus{\mathcal{D}}:w\in\Omega^{\prime}\\}\cong\mathbb{H}^{4}$
(this is immediate from Lemma 6.1, since $X\cap\partial S(4)=\emptyset$, see
the construction of $\widehat{{\mathcal{W}}}_{1}$ in proof of Lemma 7.10). If
we now restrict the homeomorphisms $f_{s}$ to the subset
$\Omega^{\prime\prime}=\\{z\in\Omega:\textrm{Re}(z)\geq
0,\textrm{Im}(z)>0\\}\cong\mathbb{H}^{2}$, then $f_{s}$ restricts to a
homeomorphism onto
$\frac{\left(\Omega^{\prime\prime}\times(0,1]\right)}{\sim}\cong\mathbb{H}^{3}$
and this is combinatorically equivalent to the homeomorphism

$f_{0}:\widehat{{\mathcal{W}}}_{1}\smallsetminus{\mathcal{D}}_{0}\rightarrow\frac{\left(\Omega^{\prime\prime}\times(0,1]\right)}{\sim}\cong\mathbb{H}^{3},$

given in Lemma 7.10, using the corresponding $I$-bundle structure. Moreover,
we can see that for every $z_{4}\in\Omega^{\prime\prime}$, the quadrilaterals
$[0,1,z_{3},z_{4}]\in\widehat{{\mathcal{W}}}_{1}$ have the property of unique
convergence of fibers as discussed in Remark 7.15, which implies that the
function

$\big{(}\Omega^{\prime\prime}\times(0,1]\big{)}\times[0,1)\rightarrow\overline{{\mathcal{W}}}_{1}\smallsetminus{\mathcal{D}},\qquad(w,t,s)\mapsto[0,1,\gamma_{\varphi_{s}(w)}(t),\varphi_{s}(w)]$

induces the homeomorphism
$Y=\\{[0,1,z,w]\in\overline{{\mathcal{W}}_{1}}\smallsetminus{\mathcal{D}}:w\in\Omega^{\prime\prime}\\}\cong\mathbb{H}^{4}$.
We can observe that $X$ and $Y$ intersect at their boundary in

$X\cap
Y=\\{[0,1,z,w]\in\overline{{\mathcal{W}}_{1}}\smallsetminus{\mathcal{D}}:\textrm{Re}(w)=0\\}\cong\mathbb{H}^{3},$

moreover, $\partial X\smallsetminus X\cap Y\cong\mathbb{H}^{3}$ and $\partial
Y\smallsetminus X\cap Y\cong\mathbb{H}^{3}$. As
$\overline{{\mathcal{W}}_{1}}\smallsetminus{\mathcal{D}}=X\cup Y$ can be
obtained as two spaces homeomorphic to $\mathbb{H}^{4}$ glued together by a
well placed copy of $\mathbb{H}^{3}$ at their boundaries, then by Theorem A.1,
we conclude
$\overline{{\mathcal{W}}_{1}}\smallsetminus{\mathcal{D}}\cong\mathbb{H}^{4}$.

Proof for the set $\overline{{\mathcal{U}}_{1}}\smallsetminus{\mathcal{D}}$.
We proceed as in the previous case by taking the function

$F:\big{(}\Omega\times(0,1]\big{)}\times(0,1)\rightarrow\overline{{\mathcal{U}}_{1}}\smallsetminus{\mathcal{D}},\qquad(w,t,s)\mapsto[0,1,\gamma_{\varphi_{s}(w)}(t),\varphi_{s}(w)],$

with $\gamma_{z}$ the fibers of the $I$-bundle on
$\textrm{cl}({\mathcal{U}}_{1})\cap h^{-1}(s)$ and $\varphi_{s}$ an
homeomorphism between $\Omega=\\{z\in\mathbb{D}^{2}:\textrm{Im}(z)\geq 0\\}$
and the base space of the bundle, cf. Figure 5. The function $F$ induces the
homeomorphism
$\textrm{cl}({\mathcal{U}}_{1})\smallsetminus{\mathcal{D}}\cong\mathbb{H}^{4}$,
but as illustrated in Example 7.14 and Remark 7.15, such function is not
continuous when approaching to the boundary at $4$-segments. We overcome this
difficulty by restricting the function to specific regions were we do have
continuity (which is everything we need to obtain cylinder structures) and
proving that $\overline{{\mathcal{U}}_{1}}\smallsetminus{\mathcal{D}}$ is a
manifold with boundary.

Just as it happened in the previous case, when the fourth vertex $z_{4}$ is
bounded away from the real line, we have uniqueness of convergence of fibers,
so the function $F$ induces a cylinder structure when we impose the condition
$\textrm{Im}(z_{4})>0$ and thus, we have
$\overline{{\mathcal{U}}_{1}}\smallsetminus(\overline{{\mathcal{V}}_{1}}\cup{\mathcal{D}})\cong\mathbb{H}^{4}$
(this set is obtained by extracting the front face in each level set
$\overline{{\mathcal{U}}_{1}}\cap h^{-1}(s)$, such faces correspond to
trapezoid quadrilaterals for $\overline{{\mathcal{U}}_{1}}\cap h^{-1}(s)$ with
$0<s<1$, and to 4-segments in the boundary $\widehat{{\mathcal{U}}}_{1}\cap
h^{-1}(0)$). We can also observe that if we restrict the convergence to
$4$-segments with only trapezoid quadrilaterals, the phenomenon of bad
convergence in Example 7.14 don’t appear, so that we have again uniqueness of
convergence of fibers and thus, the function $F$ induces the homeomorphism
$(\overline{{\mathcal{U}}_{1}}\cap\overline{{\mathcal{V}}_{1}})\smallsetminus{\mathcal{D}}\cong\mathbb{H}^{3}$.
We have thus the decomposition

$\big{(}\overline{{\mathcal{U}}_{1}}\smallsetminus{\mathcal{D}}\big{)}=\big{(}\overline{{\mathcal{U}}_{1}}\smallsetminus(\overline{{\mathcal{V}}_{1}}\cup{\mathcal{D}})\big{)}\sqcup\big{(}(\overline{{\mathcal{U}}_{1}}\cap\overline{{\mathcal{V}}_{1}})\smallsetminus{\mathcal{D}}\big{)}\cong\mathbb{H}^{4}\sqcup\mathbb{H}^{3},$

where the inclusion is
$(\overline{{\mathcal{U}}_{1}}\cap\overline{{\mathcal{V}}_{1}})\smallsetminus{\mathcal{D}}\hookrightarrow\partial\big{(}\overline{{\mathcal{U}}_{1}}\smallsetminus{\mathcal{D}}\big{)}$.
From Lemmas 5.7 and 7.10, it follows that

$\partial(\overline{{\mathcal{U}}_{1}}\smallsetminus{\mathcal{D}})=(\widehat{{\mathcal{U}}}_{1}\smallsetminus{\mathcal{D}}_{0})\cup\partial(\textrm{cl}({\mathcal{U}}_{1})\smallsetminus{\mathcal{D}})\cong\mathbb{H}^{3}\cup\big{(}\mathbb{H}^{2}\times(0,1)\big{)}\cong\mathbb{H}^{3}\cup\mathbb{H}^{3},$

and as the sets $(\widehat{{\mathcal{U}}}_{1}\smallsetminus{\mathcal{D}}_{0})$
and $\partial(\textrm{cl}({\mathcal{U}}_{1})\smallsetminus{\mathcal{D}})$ are
glued on
$\partial(\widehat{{\mathcal{U}}}_{1}\smallsetminus{\mathcal{D}}_{0})\cong\mathbb{H}^{2}$,
then we have that
$\partial(\overline{{\mathcal{U}}_{1}}\smallsetminus{\mathcal{D}})\cong\mathbb{H}^{3}$
by Theorem A.1.

Moreover, we can proof that
$\overline{{\mathcal{U}}_{1}}\smallsetminus{\mathcal{D}}$ is manifold with
boundary (by all the previous discussion, it only remains to prove this fact
around convex $4$-segments): take a $4$-segment
$[0,1,x,y]\in\overline{{\mathcal{U}}_{1}}\smallsetminus{\mathcal{D}}$ (we have
the conditions $0<x<2$, $-1<y<1$ and $y<x$), then for $0<\varepsilon<|x-y|/2$
small enough, we consider the neighborhood

$A=\\{[0,1,z_{3},z_{4}]\in\overline{{\mathcal{U}}_{1}}\smallsetminus{\mathcal{D}}:z_{3}\in
S_{\varepsilon}(x),\ z_{4}\in S_{\varepsilon}(y)\\},$

where
$S_{\varepsilon}(z)=\\{w\in\mathbb{C}:|\textrm{Re}(z-w)|<\varepsilon,|\textrm{Im}(z-w)|<\varepsilon\\}$
is a square neighborhood around the complex number $z$. We distinguish to
cases:

  1. 1.

When $0\leq x+\varepsilon\leq 1$, in which case, the neighborhood $A$ is just
the two $\varepsilon$-squares $S_{\varepsilon}(x)$ and $S_{\varepsilon}(y)$
with the extra conditions $0\leq\textrm{Im}(z_{4})$ and
$0\leq\textrm{Im}(z_{3})\leq\textrm{Im}(z_{4})$, see Figure 17.$a)$. More
precisely, we have a map

$\\{(a,b)\in\mathbb{R}^{2}:|a|,|b|<\varepsilon\\}\times\\{(c,d)\in\mathbb{R}^{2}:0\leq
c\leq d<\varepsilon\\}\rightarrow A,$

given by $\big{(}(a,b),(c,d)\big{)}\mapsto[0,1,x+a+ic,y+b+id]$, which is a
homeomorphism, so that
$A\cong\mathbb{R}^{2}\times\mathbb{H}^{2}\cong\mathbb{H}^{4}$.

  2. 2.

The general case, where the neighborhood $A$ is just the two
$\varepsilon$-squares $S_{\varepsilon}(x)$ and $S_{\varepsilon}(y)$ with the
extra conditions $0\leq\textrm{Im}(z_{4})$,
$\textrm{Im}(z_{3})\leq\textrm{Im}(z_{4})$ and $z_{3}$ is above a piecewise
linear curve (above $\\{tz_{4}+(1-t):t\in\mathbb{R}\\}$ when
$\textrm{Re}(z_{3})\geq 1$ and $\textrm{Im}(z_{3})\geq 0$ when
$\textrm{Re}(z_{3})\leq 1$), see Figure 17.$b)$. This is topologically
equivalent to the previous case, so we have once again $A\cong\mathbb{H}^{4}$.

$a)$$\bullet$$-1$$\bullet$$0$$\bullet$$1$$\bullet$$y$$\varepsilon$$\bullet$$z_{4}$$\bullet$$z_{3}$$\bullet$$x$$\varepsilon$$b)$$\bullet$$0$$\bullet$$y$$\bullet$$1$$\bullet$$x$$\bullet$$2$$\varepsilon$$\bullet$$z_{4}$$\bullet$$z_{3}$$\varepsilon$
Figure 17: Square neighborhoods around non-convex 4-segments in
$\overline{{\mathcal{U}}_{1}}\smallsetminus{\mathcal{D}}$.

By Brown’s theorem [15], there is a collar neighborhood around the set of
convex $4$-segments
$\mathbb{H}^{4}\cong(\widehat{{\mathcal{U}}}_{1}\cap\widehat{{\mathcal{V}}}_{1}\smallsetminus{\mathcal{D}}_{0})\times[0,1]^{2}\hookrightarrow\overline{{\mathcal{U}}_{1}}\smallsetminus{\mathcal{D}}$.
This collar neighborhood can be described as follows: there is a continuous
function $f:[-1,1]\rightarrow\mathbb{D}^{2}$ such that $f(-1)=f(1)=0$,
$0<f(t)<1$ for every $t\in(-1,1)$, so that the collar neighborhood is thus
just the set

$\mathcal{C}=\\{[0,1,z_{3},z_{4}]\in\overline{{\mathcal{U}}_{1}}\smallsetminus{\mathcal{D}}:\textrm{Im}(z_{4})\leq
f(\textrm{Re}(z_{4}))\\}\cong\mathbb{H}^{4}.$

This neighborhood has three boundary components defined by the identities
$\textrm{Im}(z_{3})=\textrm{Im}(z_{4})$ (corresponding to
$\overline{{\mathcal{U}}_{1}}\cap\overline{{\mathcal{V}}_{1}}$),
$\textrm{Im}(z_{3})=0$ (corresponding to $\widehat{{\mathcal{U}}}_{1}$), and
$\textrm{Im}(z_{4})=f(t)$. The complement of the collar neighborhood can be
uniformized with $F$ giving us

$\overline{{\mathcal{U}}_{1}}\smallsetminus({\mathcal{D}}\cup\mathcal{C})=\\{[0,1,z_{3},z_{4}]\in\overline{{\mathcal{U}}_{1}}\smallsetminus{\mathcal{D}}:\textrm{Im}(z_{4})\geq
f(\textrm{Re}(z_{4}))\\}\cong\mathbb{H}^{4}.$

The collar neighborhood and its complement inside
$\overline{{\mathcal{U}}_{1}}\smallsetminus{\mathcal{D}}$ intersect at

$\\{[0,1,z_{3},t+if(t)]\in\overline{{\mathcal{U}}_{1}}\smallsetminus{\mathcal{D}}:-1<t<1\\}\cong\mathbb{H}^{3},$

and thus
$\overline{{\mathcal{U}}_{1}}\smallsetminus{\mathcal{D}}\cong\mathbb{H}^{4}$
by Theorem A.1. ∎

Now we are able to give a description of the closure $\overline{S(4)}$ and of
the boundary $\partial S(4)$ as an asymptotic decomposition.

######  Theorem 7.17.

The set $\overline{S(4)}\smallsetminus{\mathcal{D}}$ has two connected
components, both homeomorphic to $\mathbb{H}^{4}$.

###### Proof.

We use Lemma 7.16 and Theorem A.1 to glue pairwise copies of $\mathbb{H}^{4}$.
The proof follows by comparing the combinatorics of each level as done in
Corollary 7.13. ∎

The combination of Theorem 7.17 and Proposition 7.9, gives the conclusion of
Theorem 1.6.

###  7.4 Compactification of the moduli space ${\mathcal{M}}(4)$

We describe the quotient $\partial{\mathcal{M}}(4):=\partial
S(4)/\langle\sigma,\tau\rangle$ and some properties of the compactification
$\overline{{\mathcal{M}}(4)}$.

######  Lemma 7.18.

If we denote by
${\mathcal{D}}^{\prime}={\mathcal{D}}/\langle\sigma,\tau\rangle$, then
${\mathcal{D}}^{\prime}\cap{\mathcal{M}}(4)\cong C(\mathbb{D}^{2})$. Moreover,
the boundary ${\mathcal{D}}^{\prime}\cap\partial{\mathcal{M}}(4)$ is also
homeomorphic to $\mathbb{D}^{2}$.

###### Proof.

For every $0<s<1$, the set
$h^{-1}(s)\cap\mathrm{cl}({\mathcal{U}}_{1})\cap\mathrm{cl}({\mathcal{W}}_{1})$
is a fundamental domain for the action of $\langle\sigma,\tau\rangle$ on
${\mathcal{D}}_{s}$, which is a topological bigon whose boundary is decompose
as a union of
$h^{-1}(s)\cap\mathrm{cl}({\mathcal{U}}_{1})\cap\mathrm{cl}({\mathcal{V}}_{1})\cap\mathrm{cl}({\mathcal{W}}_{1})$
with
$h^{-1}(s)\cap\mathrm{cl}({\mathcal{U}}_{1})\cap\mathrm{cl}({\mathcal{W}}_{1})\cap\mathrm{cl}({\mathcal{V}}_{4})\cap\mathrm{cl}({\mathcal{W}}_{4})$
(see Lemma 5.10). As the only non-trivial identification on the boundary is
given by $\tau$, acting as a folding on the curve
$h^{-1}(s)\cap\mathrm{cl}({\mathcal{U}}_{1})\cap\mathrm{cl}({\mathcal{V}}_{1})\cap\mathrm{cl}({\mathcal{W}}_{1})$,
the resulting quotient is homeomorphic to $\mathbb{D}^{2}$ (c.f. proof of
Lemma 6.3). As the fundamental domain is constructed in
$\textrm{cl}({\mathcal{U}}_{1})$, then the cone structure on
${\mathcal{D}}^{\prime}\cap{\mathcal{M}}(4)$ follows from Lemma 6.1. For the
boundary part ${\mathcal{D}}^{\prime}\cap\partial{\mathcal{M}}(4)$, we proceed
by analyzing separately the actions of $\langle\sigma,\tau\rangle$ in the sets
of non-convex 4-segments ${\mathcal{N}}$ and wedges ${\mathcal{K}}$.

Non-convex 4-segments. Since ${\mathcal{N}}=\bigcup_{j=0}^{3}\sigma^{j}T$ with
$T=\\{[0,1,t,x]\colon 0\leq t\leq x\leq 1\\}$ (see Proposition 7.3 and Figure
13), the triangle $T$ is a fundamental domain for action of the subgroup
$\langle\sigma\rangle$. The element $\tau$ acts as the reflection of $T$
through the line $\\{[0,1,1-t,t]\colon\frac{1}{2}\leq t\leq 1\\}$. Then the
quotient ${\mathcal{N}}/\langle\sigma,\tau\rangle$ is identified with the
triangle

$\\{[0,1,t,x]\colon 0\leq t\leq x\leq 1-t\\},$

whose sides are

$\\{[0,1,0,t]\colon 0\leq t\leq 1\\}\cup\left\\{[0,1,t,t]\colon 0\leq
t\leq\frac{1}{2}\right\\}\cup\left\\{[0,1,t,1-t]\colon\frac{1}{2}\leq t\leq
1\right\\}.$

Wedges. The action of $\langle\sigma\rangle$ in an element
$Z=[0,1,0,z_{4}]\in{\mathcal{K}}$ is as follows:

$\sigma(Z)=[0,1,1-z_{4},1],~{}~{}\sigma^{2}(Z)=\left[0,1,0,\frac{1}{z_{4}}\right],~{}~{}\sigma^{3}(Z)=\left[0,1,\frac{z_{4}-1}{z_{4}},1\right].$

Notice that $\sigma$ and $\sigma^{3}$ switch the spheres in
${\mathcal{K}}\cong\mathbb{S}^{2}\vee\mathbb{S}^{2}$, and $\sigma^{2}$ acts as
the function $z\mapsto\frac{1}{z}$ in the sphere $\\{[0,1,0,z_{4}]\colon
z_{4}\in\mathbb{C}\mathbb{P}^{1}\\}$ (Proposition 7.3). Moreover, the map
$\tau\sigma[0,1,0,z_{4}]=[0,1,0,\overline{z_{4}}]$ is the complex conjugate in
the same sphere. Then, we can identify the quotient
${\mathcal{K}}/\langle\sigma,\tau\rangle$ with the half disc
$\\{[0,1,0,z]\colon z\in\mathbb{D},~{}Im(z)\geq 0\\}$. We conclude that the
quotients ${\mathcal{N}}/\langle\sigma,\tau\rangle$ and
${\mathcal{K}}/\langle\sigma,\tau\rangle$ are identified through the segment
$\\{[0,1,0,t]\colon 0\leq t\leq 1\\}$ at their boundaries, resulting in a
space homeomorphic to $\mathbb{D}^{2}$, see the bottom in Figure 18$.b)$. ∎

######  Remark 7.19.

The set ${\mathcal{D}}^{\prime}$, as a compactification of the open cone
$C(\mathbb{D}^{2})$ with a disc $\mathbb{D}^{2}$ at its boundary is somewhat
involved. To illustrate the asymptotic behaviour to the boundary, we make the
following identification: if we take
$\Omega=\\{(z,s)\in\mathbb{C}\times\mathbb{R}:|z|\leq 1,\ \textrm{Im}(z)\geq
s>0\\}$, then the projection

$\textrm{cl}({\mathcal{W}}_{1})\cap\textrm{cl}({\mathcal{U}}_{1})\rightarrow\Omega\times(0,1],\qquad
Z=[0,1,w,z]\mapsto(z,h(Z))$

induces the homeomorphism
$\big{(}{\mathcal{D}}^{\prime}\cap{\mathcal{M}}(4)\big{)}\cong\Omega/{\sim}\cong
C(\mathbb{D}^{2})$ stated in Lemma 7.18, where the equivalence relation
$(z,s)\sim(z^{\prime},s)$, when $\textrm{Im}(z)=s$, is the one induced by
$\tau$ in
$\overline{{\mathcal{U}}_{1}}\cap\overline{{\mathcal{V}}_{1}}\cap\overline{{\mathcal{W}}_{1}}$
(which is analogous to a reflection), see Figure 18. Under this
identification, a sequence
$\\{[z_{n},s_{n}]\\}_{n\in\mathbb{N}}\subset\Omega/{\sim}$, satisfying
$s_{n}\rightarrow 0$ and $z_{n}\rightarrow z\in\\{w\in\mathbb{C}:|w|\leq 1,\
\textrm{Im}(w)\geq 0\\}$, converges to a quadrilateral in the quotient
${\mathcal{D}}_{0}/\langle\sigma,\tau\rangle$, according to the following
three possibilities:

  1. 1.

If $\textrm{Im}(z)>0$ or $Im(z)=0$ and $z\leq 0$, then the sequence converges
uniquely to the wedge $[0,1,0,z]$.

  2. 2.

If $z\in\mathbb{R}$ and $z>0$, then the sequence converges to a $4$-segment
$[0,1,x,z]$ with $0\leq x\leq z$. We can obtain every non-convex $4$-segment
like this by controlling the slope $s_{n}/\textrm{Im}(z_{n})$, as we have the
relation

$\lim_{n\rightarrow\infty}\frac{s_{n}}{\textrm{Im}(z_{n})}=\frac{x}{2z}.$

Such sequences were constructed in the proof of Proposition 7.9.

  3. 3.

If $z=1$, then the sequence converges to a wedge $[0,1,x,1]$ (a convex
4-segment), with $1\leq x\leq 2$, which satisfies $\tau[0,1,x,1]=[0,1,0,1-x]$.
We can obtain every wedge satisfying this condition using parallelograms
$[0,1,x+is_{n},z_{n}=a_{n}+is_{n}]\in
h^{-1}(s_{n})\cap\overline{{\mathcal{U}}_{1}}\cap\overline{{\mathcal{V}}_{1}}\cap\overline{{\mathcal{W}}_{1}}$.

$a)$$\bullet$$(1,0)$$\bullet$$(-i,0)$$\bullet$$(-1,0)$$\bullet$$(i,0)$$(i,1)$$\bullet$$\bullet$$(i,s)$$\bullet$$\bullet$$\bullet$$z$$\bullet$$z^{\prime}$$\Omega$$b)$$\bullet$$\mathfrak{R}_{4}$$\bullet$$\bullet$$\bullet$$\bullet$$\Omega/{\sim}$${\mathcal{D}}_{0}/\langle\sigma,\tau\rangle$${\mathcal{K}}$${\mathcal{N}}$
Figure 18: ${\mathcal{D}}^{\prime}$ as a compactification of the open cone
$\Omega/{\sim}\cong C(\mathbb{D}^{2})$.

######  Lemma 7.20.

For every $s\in[0,1)$, the quotient
$\big{(}h^{-1}(s)\smallsetminus{\mathcal{D}}_{s}\big{)}/\langle\sigma,\tau\rangle$
is homeomorphic to $\mathbb{R}^{3}$.

###### Proof.

Recall from the proof of Lemma 6.3 that for every $s\in(0,1)$, we have a
fundamental domain of the action of $\langle\sigma,\tau,\rangle$ on
$h^{-1}(s)\smallsetminus{\mathcal{D}}_{s}$ given by
$\big{(}\overline{{\mathcal{U}}_{1}}\smallsetminus{\mathcal{D}}\big{)}\cap
h^{-1}(s)$, together with the region

$B_{s}=\big{\\{}[0,1,z_{3},z_{4}]\in\big{(}\overline{{\mathcal{W}}_{1}}\smallsetminus{\mathcal{D}})\big{)}\cap
h^{-1}(s):\textrm{Im}(z_{3})\leq\textrm{Im}(z_{4})\big{\\}}.$

If we take the identifications on the sphere
$\partial\big{(}\overline{{\mathcal{U}}_{1}}\smallsetminus{\mathcal{D}}\big{)}\cap
h^{-1}(s)\cong\mathbb{H}^{2}$ determined by the actions of elements of the
dihedral group $\langle\sigma,\tau\rangle$, we obtain a space homeomorphic to
$\mathbb{H}^{3}$, whose boundary is obtained from the intersection
$\big{(}\overline{{\mathcal{U}}_{1}}\smallsetminus{\mathcal{D}}\big{)}\cap\overline{{\mathcal{W}}_{4}}\cap
h^{-1}(s)\cong\mathbb{R}^{2}$. If we do the analogous identifications on
$\partial B_{s}$, we obtain a space homeomorphic to $\mathbb{H}^{3}$, whose
boundary is obtained from the intersection
$\big{(}\overline{{\mathcal{W}}_{1}}\smallsetminus{\mathcal{D}}\big{)}\cap\overline{{\mathcal{V}}_{4}}\cap
h^{-1}(s)\cong\mathbb{R}^{2}$. As
$\sigma^{3}\tau(\overline{{\mathcal{W}}_{1}}\cap\overline{{\mathcal{V}}_{4}})=\overline{{\mathcal{W}}_{4}}\cap\overline{{\mathcal{U}}_{1}}$,
we conclude that after performing the identifications by elements of the
dihedral group $\langle\sigma,\tau\rangle$ in the fundamental domain, we
obtain two copies of $\mathbb{H}^{3}$ glued through their boundary which
result in a space homeomorphic to $\mathbb{R}^{3}$. By making a limit process,
and using Lemma 7.10 and Lemma 7.11, we can observe that the same proof works
for the case $h^{-1}(0)=\partial S(4)$ and the result follows. ∎

######  Theorem 7.21.

The space ${\mathcal{D}}^{\prime}={\mathcal{D}}/\langle\sigma,\tau\rangle$ is
homeomorphic to an cone $C(\mathbb{D}^{2})$ compactified with the disc
${\mathcal{D}}_{0}/\langle\sigma,\tau\rangle$. Moreover, the space
$\overline{{\mathcal{M}}(4)}\smallsetminus{\mathcal{D}}^{\prime}$ is
homeomorphic to $\mathbb{H}^{4}$.

###### Proof.

This is a consequence of taking the homeomorphisms of the previous Lemma,
together with the cylinder structure given by Lemma 7.16 and Theorem A.1 which
tells us that
$\overline{{\mathcal{M}}(4)}\smallsetminus{\mathcal{D}}^{\prime}\cong\mathbb{R}^{3}\times[0,1)\cong\mathbb{H}^{4}$.
∎

Theorem 7.21 together with the description of ${\mathcal{D}}^{\prime}$ given
in Lemma 7.18 and Remark 7.19 completes the proof of Theorem 1.7.

##  8 Conclusions and open questions

In this section we make some general conclusions about the technical steps we
had to face on the paper and include some questions which we encountered along
the way.

  1. 1.

The definition of the height function $h$, given here, generalizes immediately
for every $n\geq 3$. We can see however, that when $n\geq 5$, this function
will have multiple points were it attains its maximum and thus, it doesn’t
behave as well as when $n\leq 4$. The authors believe that the following
modification to the height function will work:

$h^{\prime}([z_{1},z_{2},z_{3},z_{4}])=\frac{\min_{1\leq j\leq
4}\\{r_{j}(Z)\\}}{\max_{1\leq j,k\leq 4}|z_{j}-z_{k}|},$

where $r_{j}(Z):=d(z_{j},{\mathcal{L}}_{j})$ with
${\mathcal{L}}_{j}:=\overline{z_{j+1}z_{j+2}}\cup\overline{z_{j+2}z_{j+3}}\cup\cdots\cup\overline{z_{j-2}z_{j-1}}$,
compare with Definition 4.1. We note that, as we are adding the length of the
diagonals $|z_{j}-z_{j+k}|$ in the denominator of $h^{\prime}$, the amount of
subsets on which we must decompose $S(n)$ so that the function $h$ behaves
well, increases considerably, compare with Definition 5.3.

  2. 2.

As explained in Lemma 5.10, the decomosition of $S(4)$ into the subsets
$\\{\overline{{\mathcal{U}}_{j}},\overline{{\mathcal{V}}_{j}},\overline{{\mathcal{W}}_{j}}\\}$
and the combinatorics of their intersections, are codified into a graph as a
flag structure, dual to the graph. A combinatorically equivalent flag
structure is constructed using cones, dual to the regular euclidean
cuboctahedron in Theorem 5.12. Is there an intrinsic relation of this
combinatorial structure to quadrilaterals? or the appearance of such a
structure is a mere coincidence.

  3. 3.

The height function $h$ is differentiable on an open, dense subset of $S(4)$
and we have a well define gradient flow there, however, this flow arrives to
the singular points in finite time. Is it possible to smooth the function, so
that it is a differentiable Morse function with a unique maximum at the
regular square? such function would induce a smooth regularizing flow on
$S(4)$. It is worth noting that Theorem 6.2 gives a continuous regularizing
flow on $S(4)$, however in a non-constructive way. The question is thus, to
obtain the results given in this paper, but in the differentiable category.

  4. 4.

As an open subspace of $\mathbb{C}\mathbb{P}^{2}$, $S(4)$ inherits the Fubini-
Study metric which is no longer complete. Moreover, as both $S(4)$ and
${\mathcal{M}}(4)$ are homeomorphic to cones over $\mathbb{S}^{3}$, we can
give the spherical metric to each level set of the height function and the
corresponding cone metrics on $S(4)$ and ${\mathcal{M}}(4)$. Is there a more
natural metric on those spaces whose geodesics capture better the
deformations, such as the Weil-Petersson metric on Teichmüller spaces? This is
a geometric version of the suavization problem of $h$ in the previous point.

  5. 5.

Is it true that a topological space $X$, with boundary homeomorphic to
$\mathbb{R}^{n-1}$ and interior homeomorphic to $\mathbb{R}^{n}$, must be
homeomorphic to $\mathbb{H}^{n}$? This assertion would have made the proof of
Lemma 7.16 shorter. What if we add the condition on $X$ to be a manifold with
boundary?

  6. 6.

As exemplified in Example 7.14 and explained in Remark 7.15, the $I$-bundle
structures described in the cones
$\\{\overline{{\mathcal{U}}_{j}},\overline{{\mathcal{V}}_{j}},\overline{{\mathcal{W}}_{j}}\\}$
on each level set $h^{-1}(s)$ don’t converge good to the $I$-bundle structure
described in the cones at the boundary $h^{-1}(0)$. We overcame this
difficulty in the proof of Lemma 7.16 by restricting the $I$-bundle structures
to subsets where we had good convergence and then gluing the structures with
Theorem A.1. The conclusion of Lemma 7.16, tells us indirectly that there
exist $I$-bundle structures which do have good convergence phenomena. How can
we describe such $I$-bundle structures?

  7. 7.

If we consider the normalized side lengths of a quadrilateral

$\ell_{i}:S(4)\rightarrow\mathbb{R},\qquad\ell_{i}(Z)=\frac{2|z_{i}-z_{i+1}|}{\sum_{j=1}^{4}|z_{j}-z_{j+1}|},$

then the function $f:S(4)\rightarrow\mathbb{R}^{4}$,
$f(Z)=(\ell_{1}(Z),\ell_{2}(Z),\ell_{3}(Z),\ell_{4}(Z))$ was considered in
[19] and they proved that the fibers $f^{-1}(p)$ are one dimensional curves.
It is possible to see that the level sets of the height function $h^{-1}(s)$
are transversal to those curves, and thus, the fibers of $f$ define a singular
foliation of curves on $S(4)$. As an example of this, the curve
$f^{-1}(1/2,1/2,1/2,1/2)$ is the fiber which contains the regular square and
is defined by the intersection (see Remark 5.8.III)

$\bigcap_{j=1}^{4}\big{(}\textrm{cl}({\mathcal{U}}_{j})\cap\textrm{cl}({\mathcal{V}}_{j})\cap\textrm{cl}({\mathcal{W}}_{j})\big{)}.$

  8. 8.

The cone structure $S(4)$ give us the homeomorphisms $h^{-1}(0,1]\cong
h^{-1}(\varepsilon,1]\cong S(4)$, for some $\varepsilon>0$ and thus, it allows
us to consider the compactification $\overline{S(4)}^{eucl}\cong
h^{-1}[\varepsilon,1]$, which is homeomorphic to a closed cone over
$\mathbb{S}^{3}$. This compactification is invariant under the action of the
dihedral group, so it gives a compactification of the moduli space
$\overline{{\mathcal{M}}(4)}^{eucl}$ as a closed cone over $\mathbb{S}^{3}$ as
well and in particular
$\overline{{\mathcal{M}}(4)}^{eucl}\cong\mathbb{D}^{4}$. A reason to prefer
the compactifications of $S(4)$ and ${\mathcal{M}}(4)$ considered in this work
is because it is compatible with the spaces considered in [19], but there is
still the question on what properties capture each compactification and why
should we prefer one or the other.

  9. 9.

How much can we weaken the hypotheses of Proposition B.1? As a more concrete
example of this question, we can ask the following: If $G=N\rtimes H$ is a
group acting by homeomorphisms on a topological space $X$, such that the
action of $N$ descends to a well defined action on $(X/H)$, is it true that
$(X/H)/N$ is homeomorphic to $X/G$? See Remark B.3 and Example B.2. We should
emphasize that we do not have a reference for Remark B.3 as it is considered a
folklore result.

##  Appendix A Gluing closed balls and closed half-spaces

We prove two topological results that are used many times along the paper. The
ideas of the proofs of the two results in this appendix are extracted from
[16]. Recall the notations of the closed ball
$\mathbb{D}^{n}=\\{X\in\mathbb{R}^{n}\colon|X|\leq 1\\}$ and the upper half-
space $\mathbb{H}^{n}=\\{(x_{1},x_{2},\dots,x_{n})\in\mathbb{R}^{n}\colon
x_{n}\geq 0\\}$.

######  Theorem A.1.

Let $H_{i}^{n}\cong\mathbb{H}^{n}$ be two copies of the upper half $n$-space,
and $H_{i}^{n-1}\subset\partial H_{i}^{n}$, such that
$H_{i}^{n-1}\cong\mathbb{H}^{n-1}$ and $\overline{\partial
H_{i}^{n}\smallsetminus H_{i}^{n-1}}\cong\mathbb{H}^{n-1}$. For every
homeomorphism $\varphi\colon H_{1}^{n-1}\to H_{2}^{n-1}$, the quotient
$(H_{1}^{n}\cup H_{2}^{n})\big{/}\varphi$ is homeomorphic to $\mathbb{H}^{n}$.

###### Proof.

We will use that $\mathbb{H}^{n}=\mathbb{R}^{n-1}\times[0,\infty)$ and the
embedding $\mathbb{H}^{n-1}\hookrightarrow\mathbb{H}^{n}$ given by
$(x_{1},\dots,x_{n-1})\mapsto(x_{1},\dots,x_{n-1},0)$. The key property of the
proof is that every homeomorphism $g\colon\mathbb{R}^{n-1}\to\mathbb{R}^{n-1}$
can be extended to a homeomorphism $G\colon\mathbb{H}^{n}\to\mathbb{H}^{n}$ as
follows

$G(p,t)=(g(p),t)\qquad\forall~{}~{}p\in\mathbb{R}^{n-1}=\partial\mathbb{H}^{n},~{}~{}\text{and}~{}~{}t\in[0,\infty).$

By applying the key property twice to a homeomorphism $f\colon
H_{1}^{n-1}\to\mathbb{H}^{n-1}$, first extending to $\partial H_{1}^{n}$ and
then to the whole $H_{1}^{n}$, we obtain a homeomorphism $F\colon
H_{1}^{n}\to\mathbb{H}^{n}$ whose restriction to $H_{1}^{n-1}$ is $f$. The
homeomorphism $F\sqcup F\colon H_{1}^{n}\sqcup
H_{1}^{n}\to\mathbb{H}^{n}\sqcup\mathbb{H}^{n}$, induces a homeomorphism

$\frac{H_{1}^{n}\sqcup
H_{1}^{n}}{id_{H_{1}^{n-1}}}\cong\frac{\mathbb{H}^{n}\sqcup\mathbb{H}^{n}}{id_{\mathbb{H}^{n-1}}}\cong\mathbb{H}^{n}.$

By applying now, the key property twice to $\varphi\colon H_{1}^{n-1}\to
H_{2}^{n-1}$, we obtain a homeomorphism $\Phi\colon H_{1}^{n}\to H_{2}^{n}$
that extends $\varphi$. The homeomorphism $Id\sqcup\Phi\colon H_{1}^{n}\sqcup
H_{1}^{n}\to H_{1}^{n}\sqcup H_{2}^{n}$, induces a homeomorphism from
$(H_{1}^{n}\sqcup H_{1}^{n})\big{/}id_{H_{1}^{n-1}}$ to $(H_{1}^{n}\sqcup
H_{2}^{n})\big{/}\varphi$ and the result follows. ∎

######  Theorem A.2.

Let $D_{i}^{n}\cong\mathbb{D}^{n}$ be two copies of the closed $n$-ball, and
$D_{i}^{n-1}\subset\partial D_{i}^{n}$ such that
$D_{i}^{n-1}\cong\mathbb{D}^{n-1}$ and $\overline{\partial
D_{i}^{n}\smallsetminus D_{i}^{n-1}}\cong\mathbb{D}^{n-1}$. For every
homeomorphism $\varphi\colon D_{1}^{n-1}\to D_{2}^{n-1}$, the quotient
$(D_{1}^{n}\cup D_{2}^{n})\big{/}\varphi$ is homeomorphic to $\mathbb{D}^{n}$.

###### Proof.

The proof is analogous to the proof of Theorem A.1. We just have to use the
cone structure $\mathbb{D}^{n}\cong C(\mathbb{S}^{n-1})$ and the embedding
$\mathbb{D}^{n-1}\hookrightarrow\mathbb{D}^{n}$ given by

$(x_{1},\dots,x_{n-1})\mapsto\left(x_{1},\dots,x_{n-1},\sqrt{1-x_{1}^{2}-\cdots-
x_{n-1}^{2}~{}}\right),$

to get the property that every homeomorphism
$g\colon\partial\mathbb{D}^{n}\rightarrow\partial\mathbb{D}^{n}$ extends to a
homeomorphism of $\mathbb{D}^{n}$ as

$G\colon\mathbb{D}^{n}\rightarrow\mathbb{D}^{n},\quad
G(tp)=tg(p),\quad\forall~{}~{}p\in\mathbb{S}^{n-1}=\partial\mathbb{D}^{n}.$

∎

######  Remark A.3.

One way to obtain the hypothesis in Theorem A.2 that the complement of
$D_{i}^{n-1}$ in $\partial D_{i}^{n}$ is again a ball is by adding PL-
structures to the balls, as it is done for example in [16, Corollary 3.13].
Another way to check the hypothesis is to ask for the balls $D_{i}^{n-1}$ to
be bi-collared in $\partial D_{i}^{n}$, so that the complement ball hypothesis
follows from the solution of the Schoenflies conjecture by Brown in [14].

##  Appendix B Intermediate quotients by normal subgroups

In this appendix, we give a proof of the following result which the authors
consider is a well known result, but couldn’t find a reference for it.

######  Proposition B.1.

If $G$ is a group acting on the right by homeomorphisms on a topological space
$X$ and $N\leq G$ is normal subgroup, then there is a homeomorphism between
the double quotient $(X/N)/(G/N)$ and $X/G$.

###### Proof.

Observe first that the projection map $\pi:X\rightarrow X/G$ descends to the
quotient map $\widehat{\pi}:X/N\rightarrow X/G$. As $N$ is a normal subgroup
of $G$, the $G$-action on $X$ induces a $G$-action on $X/N$ with $N$ acting
trivially, so that it induces a $G/N$-action on $X/N$. As $\widehat{\pi}$ is
$G/N$-invariant, it induces a continuous and bijective map
$\widehat{\widehat{\pi}}:(X/N)/(G/N)\rightarrow X/G$. If
$\widehat{\widehat{A}}\subset(X/N)/(G/N)$ is any subset, then there is a
subset $A\subset X$ such that $\widehat{\widehat{A}}=\\{(xN)(G/N):x\in A\\}$,
which means that
$\widehat{\widehat{A}}=\\{\widehat{x}(G/N):\widehat{x}\in\widehat{A}\\}$,
where $\widehat{A}=\\{yN:y\in A\\}$. By definition, $\widehat{\widehat{A}}$ is
open in $(X/N)/(G/N)$ if and only if its inverse image under the successive
projection maps is open, that is, if the set $\\{(xg):x\in A,g\in G\\}=AG$ is
open. As the projection $\pi$ is an open map,
$\pi(AG)=\pi(A)=\widehat{\widehat{\pi}}(\widehat{\widehat{A}})$ is open in
$X/G$ and $\widehat{\widehat{\pi}}$ is an homeomorphism. ∎

######  Example B.2 (non-example).

Consider the dihedral group
$D_{4}=\langle\sigma\rangle\rtimes\langle\tau\rangle$ acting in the circle
$\mathbb{S}^{1}\subset\mathbb{C}$, as $\sigma(z)=e^{\pi i/2}z$ and
$\tau(z)=\overline{z}$. The quotient $\mathbb{S}^{1}/\langle\tau\rangle$ is
homeomorphic to $\\{z\in\mathbb{S}^{1}:\textrm{Im}(z)\geq 0\\}$ and the
homeomorphism $\sigma$ descends to a continuous map on
$\mathbb{S}^{1}/\langle\tau\rangle$, defined under this identification by

$\tilde{\sigma}(z)=\left\\{\begin{array}[]{lcr}e^{\pi
i/2}z,&&\textrm{Re}(z)\geq 0,\\\ e^{3\pi i/2}\overline{z},&&\textrm{Re}(z)\leq
0.\end{array}\right.$

Under the equivalence class $z\sim\tilde{\sigma}(z)$, we have an homeomorphism

$(\mathbb{S}^{1}/\langle\tau\rangle)/\tilde{\sigma}=(\mathbb{S}^{1}/\langle\tau\rangle)/\sim\cong\mathbb{S}^{1}/D_{4}.$

In this case however, the map $\tilde{\sigma}^{k}$ is non-injective, and thus,
the action of the cyclic group $\langle\sigma\rangle$ on $\mathbb{S}^{1}$
doen’t descend to a well defined action on
$\mathbb{S}^{1}/\langle\tau\rangle$.

######  Remark B.3.

As we saw in Example B.2, there are cases where the condition of the subgroup
on Proposition B.1 to be normal is no longer true and nevertheless the result
holds. This is not always the case as there examples of two non-commuting
homeomorphisms $f_{1},f_{2}:X\rightarrow X$, such that for $X_{j}=X/\langle
f_{j}\rangle$ the quotient $X_{1}/\langle f_{2}\rangle$ is not homeomorphic to
$X_{2}/\langle f_{1}\rangle$ and both spaces are not homeomorphic to
$X/\langle f_{1},f_{2}\rangle$.

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Ahtziri González, Facultad de Ingeniería Eléctrica, Universidad Michoacana de
San Nicolás de Hidalgo, Edificio Omega, Ciudad Universitaria, C.P. 58040,
Morelia, Michoacán, México. e-mail<EMAIL_ADDRESS>

Manuel Sedano-Mendoza, e-mail<EMAIL_ADDRESS>


# The Discrete Spherical Maximal Function:
A new proof of $\ell^{2}$-boundedness

Neil Lyall Ákos Magyar Alex Newman Peter Woolfitt  Department of Mathematics,
The University of Georgia, Athens, GA 30602, USA<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We provide a new direct proof of the $\ell^{2}$-boundedness of the Discrete
Spherical Maximal Function that neither relies on abstract transference
theorems (and hence Stein’s Spherical Maximal Function Theorem) nor on
delicate asymptotics for the Fourier transform of discrete spheres.

###### 2010 Mathematics Subject Classification:

42B25

The first and second authors were partially supported by grants NSF-DMS
1702411 and NSF-DMS 1600840, respectively.

##  1\. Introduction

The study of discrete analogues of central constructs of Euclidean harmonic
analysis, initiated by Bourgain [1], has grown into a vast, active area of
research. An important result in this development is the
$\ell^{p}$-boundedness of the so-called discrete spherical maximal function
[5].

Beyond its own intrinsic interest, this operator, or more precisely certain
“mollified variants”, play a crucial role in studying certain geometric point
configurations in positive density subsets of the integer lattice, see [2].

Let $d\geq 5,\,\lambda^{2}\in\mathbb{N}$, and
$N_{\lambda}:=|\\{m\in\mathbb{Z}^{d}:\ |m|=\lambda\\}|.$ It is well-known, see
for example [7], that

$c_{d}\lambda^{d-2}\leq N_{\lambda}\leq C_{d}\lambda^{d-2}$

for some constants $0<c_{d}<C_{d}$. For $f:\mathbb{Z}^{d}\to\mathbb{R}$ define
the discrete spherical averages

$A_{\lambda}f(n)=N_{\lambda}^{-1}\sum_{|m|=\lambda}f(n-m)$

and the maximal operator

$A_{\ast}f(n)=\sup_{\lambda}|A_{\lambda}f(n)|.$

The variables $n,m$ in the two equations above, and throughout this short
note, are always assumed to be in $\mathbb{Z}^{d}$. Furthermore, the parameter
$\lambda$ will always be assumed to satisfy $\lambda^{2}\in\mathbb{N}$.

In [5] it was shown that for $p>d/(d-2)$ one has the estimate

$\|A_{\ast}f\|_{p}\leq C_{p,d}\,\|f\|_{p}$

where $\|f\|_{p}=(\sum_{x}|f(x)|^{p})^{1/p}$ denotes the
$\ell^{p}(\mathbb{Z}^{d})$ norm of the function $f$. It was further noted in
[5] that the condition that $d\geq 5$ and $p>d/(d-2)$ are both sharp.

The approach taken in [5] had three main steps. The first step was to
approximate $A_{\lambda}$ by an infinite sum of simpler operators
$M_{\lambda}^{a/q}$, each associated to a reduced fraction $a/q$, with
$0<a/q\leq 1$. A general abstract transference theorem, which allows one to
pass from certain convolution operators on $\mathbb{R}^{d}$ to analogous
operators on $\mathbb{Z}^{d}$, was then used to analyze each
$M_{\lambda}^{a/q}$. In particular, this approach makes use of Stein’s
Spherical Maximal Function Theorem [6]. The final step of the argument is to
show that the approximation taken in the first step is adequate, this step
uses the full asymptotic expansion for the Fourier transform of (the indicator
function of) the discrete sphere of radius $\lambda$ in $\mathbb{Z}^{d}$.

In this note we provide a short direct proof of the $\ell^{2}$ case of the
main result in [5]. Our direct proof relies on the observation that one
obtains gains in $\ell^{2}$ for maximal operators at a single dyadic scale,
when applied to functions whose Fourier transform is suitably localized away
from rational points with suitably small denominators, specifically
Proposition 1 below. This combined with an almost orthogaonality argument
quickly leads to the proof Theorem 1 below. Note that we do not need the full
asymptotic expansion of the underlying multipliers neither any transference
arguments to utilise Stein’s spherical maximal theorem.

Our main result is the following,

######  Theorem 1.

If $d\geq 5$, then

$\|A_{\ast}f\|_{2}\leq C_{d}\,\|f\|_{2}.$

##  2\. Key estimates for maximal operators at a single dyadic scale

Recall that for $f\in\ell^{1}(\mathbb{Z}^{d})$ we define its Fourier transform
$\widehat{f}:\mathbb{T}^{d}\to\mathbb{C}$ by

$\widehat{f}(\alpha)=\sum_{n\in\mathbb{Z}^{d}}f(n)e^{-2\pi in\cdot\alpha}.$

Before stating Proposition 1 we need to introduce some additional notation.
For any integer $j\geq 0$ we let
$q_{j}=\operatorname{lcm}\\{1,2,\dots,2^{j}\\}$ and note that $q_{j}\asymp
e^{2^{j}}$. For any non-negative integers $j$ and $k$ that satisfy $2^{j}\leq
k$ , we let

(1)
$\Omega_{j,k}:=\\{\alpha\in\mathbb{T}^{d}:\alpha\in[-2^{j-k},2^{j-k}]^{d}+(q_{j}^{-1}\mathbb{Z})^{d}\\}.$

######  Proposition 1.

If $d\geq 5$, $k\in\mathbb{N}$, and $1\leq j\leq\log_{2}(k)-2$, then one has
the estimate

(2) $\Bigl{\|}\sup_{2^{k}\leq\lambda\leq
2^{k+1}}|A_{\lambda}f|\Bigr{\|}_{2}\ll 2^{-j/2}j^{-1}\|f\|_{2}$

whenever $\operatorname{supp}\widehat{f}\subseteq\Omega_{j,k}^{c}$, where
$\Omega_{j,k}^{c}$ denotes the complement of $\Omega_{j,k}$.

In the Proposition above, and for the rest of this short note, we use the
notation $A\ll B$ to denote that $A\leq CB$ for some constant $C$ that may
depend on $d$, which we consider fixed and greater than or equal to 5.

The proof of Proposition 1 is presented in Section 4, while the reduction of
Theorem 1 to Proposition 1 is presented in Section 3 below. We conclude this
section by noting that Proposition 1 immediately implies the following
“mollified variant” of Theorem 1 which is of independent interest.

######  Theorem 2.

If $d\geq 5$, $\eta>0$, and $L\geq q_{\eta}^{4}$, then one has the estimate

(3)
$\Bigl{\|}\sup_{\lambda\geq\eta^{-2}L}|A_{\lambda}f|\Bigr{\|}_{2}\ll\eta\,\|f\|_{2}$

whenever $\operatorname{supp}\widehat{f}\subseteq\Omega_{\eta,L}^{c}$, with
$\Omega_{\eta,L}=\\{\alpha\in\mathbb{T}^{d}:\alpha\in[-L^{-1},L^{-1}]^{d}+(q_{\eta}^{-1}\mathbb{Z})^{d}\\}$
and $q_{\eta}=\operatorname{lcm}\\{1\leq q\leq\eta^{-2}\\}$.

Indeed, note that in proving (3) one may restrict the sup to
$\eta^{-2}L\leq\lambda\leq 2\eta^{-2}L$. Choosing $k,j\in\mathbb{N}$ such that
$2^{k}\leq\eta^{-2}L\leq 2^{k+1}$ and $2^{j}\geq\eta^{-2}$ we have that
$2^{k-j}\leq L$ and hence $\Omega_{j,k}\subseteq\Omega_{\eta,L}$. Applying
Proposition 1 with $j$ and $k$ chosen as above implies Theorem 2.

This provides a slight strengthening of Proposition 5 in [2], more importantly
it provides a significantly simpler direct proof.

##  3\. Proof of Theorem 1

###  3.1. A smooth sampling function supported on $\Omega_{j,k}$

Let $\psi\in\mathcal{S}(\mathbb{R}^{d})$ be a Schwartz function satisfying

$1_{Q}(\xi)\leq\widetilde{\psi}(\xi)\leq 1_{2Q}(\xi)$

where $Q=[-1/2,1/2]^{d}$ and

$\widetilde{\psi}(\xi):=\int_{\mathbb{R}^{d}}\psi(x)e^{-2\pi ix\cdot\xi}dx$

denote the Fourier transform of $\psi$ on $\mathbb{R}^{d}$. For a given
$q\in\mathbb{N}$ and $L>q$ we define $\psi_{q,L}:\mathbb{Z}^{d}\to\mathbb{R}$
as

$\psi_{q,L}(m)=\begin{cases}\left(\frac{q}{L}\right)^{d}\psi\left(\frac{m}{L}\right)&\textrm{if
}m\in(q\mathbb{Z})^{d}\\\ 0&\textrm{otherwise}\end{cases}$

Writing $m=qr+s$ with $r\in\mathbb{Z}^{d}$ and
$s\in\mathbb{Z}^{d}/q\mathbb{Z}^{d}$, it follows from Poisson summation that

$\widehat{\psi}_{q,L}(\alpha)=\sum_{m\in\mathbb{Z}^{d}}\psi(m)e^{-2\pi
im\cdot\alpha}$

is a $q^{-1}$-periodic function on $\mathbb{T}^{d}$ that satisfies

$\widehat{\psi}_{q,L}(\alpha)=\sum_{\ell\in\mathbb{Z}^{d}}\widetilde{\psi}(L(\alpha-\ell/q)).$

For a given $k\in\mathbb{N}$ and $0\leq j\leq J_{k}:=[\log_{2}(k)]-2$, we now
define the sampling function

(4) $\Psi_{j,k}=\psi_{q_{j},2^{k-j}}$

and note that
$\operatorname{supp}{\widehat{\Psi}_{j,k}}\subseteq\Omega_{j,k}$.

Finally we define $\Delta\Psi_{j,k}=\Psi_{j+1,k}-\Psi_{j,k}$ and note the
important almost orthogonality property they enjoy.

######  Lemma 1.

There exists a constant $C=C_{\Psi}>0$ such that

$\sum_{k\geq 2^{j}}|\widehat{\Delta\Psi}_{k,j}(\alpha)|^{2}\leq C$

uniformly in $j\in\mathbb{N}$ and $\alpha\in\mathbb{T}^{d}.$

###### Proof of Lemma 1.

Note that $\Omega_{k+1,j}\subseteq\Omega_{k,j}.$ Now fix $j\in\mathbb{N}$. If
$\alpha\notin\Omega_{2^{j},j}$, then $\widehat{\Delta\Psi}_{k,j}(\alpha)=0$.

If $\alpha\in\Omega_{2^{j},j}$, then we define $k_{1}=k_{1}(j):=\max\\{k\geq
2^{j}:\alpha\in\Omega_{k,j}\\}$. Then there exists a unique
$\ell_{1}\in\mathbb{Z}^{d}$ such that $|\alpha-\ell_{1}/q_{j}|\leq
2^{j-k_{1}}.$ Clearly $\widehat{\Delta\Psi}_{k,j}(\alpha)=0$ if $k>k_{1}$,
while if $2^{j}\leq k\leq k_{1}$ we have
$\widehat{\Psi}_{k,j}(\alpha)=\tilde{\Psi}(2^{k-j}(\alpha-\ell_{1}/q_{j}))$.
It therefore follows, by writing
$\Delta\Psi_{k,j}=(\Psi_{k,j+1}-1)+(1-\Psi_{k,j})$, that

$|\widehat{\Delta\Psi}_{k,j}(\alpha)|\leq
C_{\Psi}\,2^{k-j}|\alpha-\ell_{1}/q_{j}|\leq C_{\Psi}\,2^{k-k_{1}}$

and hence that

$\sum_{k\geq 2^{j}}|\widehat{\Delta\Psi}_{k,j}(\alpha)|^{2}\leq
C_{\Psi}\,\sum_{1\leq k\leq
k_{1}}2^{-2(k_{1}-k)}\leq\frac{4}{3}\,C_{\Psi}.\qed$

###  3.2. Proof that Proposition 1 implies Theorem 1

Let

(5) $M_{k}f:=\sup_{2^{k}\leq\lambda\leq 2^{k+1}}|A_{\lambda}f|.$

Writing

$f=f*\Psi_{k,0}+\sum_{j=0}^{J_{k}-1}f*\Delta\Psi_{k,j}+(f-f*\Psi_{k,J_{k}})$

it follows by subadditivity that

(6) $M_{k}f\leq
M_{k}(f*\Psi_{k,0})+\sum_{j=0}^{J_{k}-1}M_{k}(f*\Delta\Psi_{k,j})+M_{k}(f-f*\Psi_{k,J_{k}})$

Theorem 1 will now follow from a few observations and applications of
Proposition 1, in light of the fact that

$A_{*}f=\sup_{k}M_{k}f.$

First we note that it is straightforward to verify that the first term on the
right in (6) above satisfies

$M_{k}(f*\Psi_{k,0})\leq C_{\Psi}\mathcal{H}f$

uniformly in $k$, where

$\mathcal{H}f(n)=\sup_{\ell>0}\frac{1}{(2\cdot
2^{\ell}+1)^{d}}\Bigl{|}\sum_{m\in[-2^{\ell},2^{\ell}]^{d}\cap\mathbb{Z}^{d}}f(n-m)\Bigr{|}$

denotes the discrete Hardy-Littlewood maximal operator. Since, by the same
arguments as in Euclidean spaces, we have $\|\mathcal{H}f\|_{2}\ll\|f\|_{2}$,
it follows that

$\sup_{k}\|M_{k}(f*\Psi_{k,0})\|_{2}\ll\|f\|_{2}.$

For the middle terms in (6) we first note that

$\sup_{k}\sum_{j=0}^{J_{k}-1}M_{k}(f*\Delta\Psi_{k,j})\ll\Bigl{(}\sum_{k=0}^{\infty}\Bigl{|}\sum_{j=0}^{J_{k}-1}M_{k}(f*\Delta\Psi_{k,j})\Bigr{|}^{2}\Bigr{)}^{1/2}$

Taking $\ell^{2}$ norms of both sides of the inequality above and applying
Minkowski’s inequality, followed by an application of Proposition 1, gives

$\displaystyle\Bigl{\|}\sup_{k}\sum_{0\leq j\leq
J_{k}}M_{k}(f*\Delta\Psi_{k,j})\Bigr{\|}_{2}$
$\displaystyle\leq\sum_{j}\Bigl{(}\sum_{k\geq
2^{j}}\|M_{k}(f*\Delta\Psi_{k,j})\|_{2}^{2}\Bigr{)}^{1/2}$
$\displaystyle\ll\sum_{j}2^{-j/2}\Bigl{(}\sum_{k\geq
2^{j}}\|f*\Delta\Psi_{k,j}\|^{2}_{2}\Bigr{)}^{1/2}\ll\|f\|_{2}$

where the last inequality above follows from Lemma 1.

One more application of Proposition 1 with $j=[\log_{2}k]-2$ to the last term
in (6) gives

$\Bigl{\|}\sup_{k}M_{k}(f-f*\Psi_{k,J_{k}})\Bigr{\|}_{2}\leq\Bigl{(}\sum_{k=1}^{\infty}\|M_{k}(f-f*\Psi_{k,J_{k}})\|_{2}^{2}\Bigr{)}^{1/2}\ll\Bigl{(}\sum_{k=1}^{\infty}k^{-1}(\log_{2}k)^{-2}\Bigr{)}^{1/2}\|f\|_{2}\ll\|f\|_{2}.\qed$

##  4\. Proof of Proposition 1

Fix $\varepsilon=2^{-2k}$. We start by observing that

$1_{\\{|m|=\lambda\\}}(m)=\int_{0}^{1}e^{2\pi
i(|m|^{2}-\lambda^{2})t}dt=e^{2\pi\varepsilon\lambda^{2}}\int_{0}^{1}e^{2\pi
i|m|^{2}(t+i\varepsilon)}e^{-2\pi i\lambda^{2}t}\,dt$

where $1_{\\{|m|=\lambda\\}}$ denotes the indicator function of the discrete
sphere of radius $\lambda$ in $\mathbb{Z}^{d}$.

Since $N_{\lambda}\asymp\lambda^{d-2}$ it therefore follows that

$M_{k}f\ll\sup_{2^{k}\leq\lambda\leq
2^{k+1}}\frac{1}{\lambda^{d-2}}\int_{0}^{1}|f*s_{t}|\,dt$

where $s_{t}(m)=e^{2\pi i|m|^{2}(t+i\varepsilon)}$, and hence that

$\|M_{k}f\|_{2}\ll\varepsilon^{(d-2)/2}\int_{0}^{1}\|f*s_{t}\|_{2}\,dt\leq\varepsilon^{(d-2)/2}\left(\int_{0}^{1}\|\widehat{s_{t}}\,1_{\Omega^{c}_{j,k}}\|_{\infty}\,dt\right)\|f\|_{2}.$

Thus, in order to prove Proposition 1 it suffices to show that

(7)
$\int_{0}^{1}\|\widehat{s_{t}}\,1_{\Omega^{c}_{j,k}}\|_{\infty}\,dt\ll\varepsilon^{-(d-2)/2}2^{-j/2}j^{-1}.$

To do this we will employ the circle method and decompose the interval into
Farey arcs, that is neighborhoods $V_{a,q}$ of reduced rationals $a/q$ which
allows us to estimate $\widehat{s}_{t}(\xi)$ by using Poisson summation and
properties of Gaussian sums. Specifically, we decompose the interval $[0,1]$
into neighborhoods of rationals whose denominator is smaller than $2^{k}$ as
follows: Let

$H=\\{a/q:1\leq q\leq 2^{k},0<a\leq q,(a,q)=1\\}$

and define

$V_{a,q}=\left\\{t\in[0,1]:\left|t-a/q\right|=\min_{r\in H}|t-r|\right\\}.$

Note that, by Dirichlet’s principle, for every $t\in[0,1]$, there exists
$a/q\in H$ such that $|t-a/q|\leq 2^{-k}q^{-1}$, thus we have that
$|V_{a,q}|\leq 2^{-k+1}q^{-1}$. Also, if $a/q\neq a^{\prime}/q^{\prime}$ with
$(a,q)=(a^{\prime},q^{\prime})=1$ and $1\leq q,q^{\prime}\leq 2^{k}$ then
$|a/q-a^{\prime}/q^{\prime}|\geq 1/(qq^{\prime})\geq 2^{-k}q^{-1}$, hence
$|V_{a,q}|\geq 2^{-k}q^{-1}$. Thus the Farey arcs $V_{a,q}$ at level $2^{k}$
provide a partition (up to endpoints) of [0,1] into intervals of length
$|V_{a,q}|\approx 2^{-k}q^{-1}$.

It follows from Poisson summation that for $1\leq a\leq q$, $(a,q)=1$, $1\leq
q\leq 2^{k}$ one has

(8) $|\widehat{s_{t}}(\alpha)|\leq
q^{-d/2}(\varepsilon+|\tau|)^{-d/2}\sum_{\ell\in\mathbb{Z}^{d}}e^{-\frac{\pi}{2}|\alpha-\ell/q|^{2}/({\varepsilon+\varepsilon^{-1}|\tau|^{2}})}$

for each $t\in V_{a,q}$ with $\tau=t-a/q$. The details of the calculation to
derive estimate (8) are laid out more carefully in [3], but they can be
briefly summarize as follows: First write $\widehat{s_{t}}$ as a product of
one dimensional functions. An application of Poisson summation and a change of
variables leaves a double sum that can be recognized as a quadratic Gaussian
sum, which can be bounded by $q^{-1/2}$, and a sum of terms involving
$\widetilde{s}_{\tau}(\ell/q-\alpha)$ which has a simple closed form. See
formula $(12)$ in [3].

Since $|\tau|\leq 2^{-k}q^{-1}$, it follows that
$q^{2}(\varepsilon+\varepsilon^{-1}|\tau|^{2})\ll 1$, and hence that

$\sum_{\ell\in\mathbb{Z}^{d}}e^{-\frac{\pi}{2}|\alpha-\ell/q|^{2}/({\varepsilon+\varepsilon^{-1}|\tau|^{2}})}\ll
1$

which in turn implies that if $t\in V_{a,q}$ with $t=a/q+\tau$, then

(9) $\|\widehat{s_{t}}\|_{\infty}\ll q^{-d/2}(\varepsilon+|\tau|)^{-d/2}.$

We write

(10)
$\int_{0}^{1}\|\widehat{s_{t}}\chi_{{}_{\Omega^{c}_{j,k}}}\|_{\infty}\,dt\,=\,\sum_{q|q_{j}}\sum_{(a,q)=1}\int_{V_{a,q}}\|\widehat{s_{t}}\,1_{\Omega^{c}_{j,k}}\|_{\infty}dt\
+\ \sum_{q\nmid
q_{j}}\sum_{(a,q)=1}\int_{V_{a,q}}\|\widehat{s_{t}}\,1_{\Omega^{c}_{j,k}}\|_{\infty}\,dt.$

In order to estimate the first double sum above we consider separately the
case when $|\tau|\geq 2^{j/2}\varepsilon$ and $|\tau|\leq 2^{j/2}\varepsilon$.
When $|\tau|\geq 2^{j/2}\varepsilon$ we use estimate (9) to bound it by

(11)
$\displaystyle\sum_{q|q_{j}}\sum_{(a,q)=1}\int_{V_{a,q}}q^{-d/2}(\varepsilon+|\tau|)^{-d/2}dt$
$\displaystyle\ll\sum_{q|q_{j}}q^{-d/2+1}\int_{\varepsilon
2^{j/2}}^{\infty}(\varepsilon+|\tau|)^{-d/2}d\tau$
$\displaystyle\ll\varepsilon^{-(d-2)/2}2^{-j(d-2)/4}\,\sum_{q|q_{j}}q^{-d/2+1}.$

When $|\tau|\leq 2^{j/2}\varepsilon$ we note that because $q|q_{j}$ we have
that

(12)
$\sum_{\ell\in\mathbb{Z}^{d}}e^{-\frac{\pi}{2}|\alpha-\ell/q|^{2}/(\varepsilon+\varepsilon^{-1}|\tau|^{2})}=e^{-\frac{\pi}{2}|\alpha-\ell_{0}/q|^{2}/(\varepsilon+\varepsilon^{-1}|\tau|^{2})}+\sum_{\ell\neq\ell_{0}}e^{-\frac{\pi}{2}|\alpha-\ell/q|^{2}/(\varepsilon+\varepsilon^{-1}|\tau|^{2})}$

where $\ell_{0}$ denotes the nearest integer to $q\alpha$. For every
$\alpha\in\Omega^{c}_{j,k}$ we have
$|\alpha-\frac{\ell_{0}}{q}|=|\alpha-\frac{q_{j}\ell_{0}/q}{q_{j}}|\geq
2^{j-k}$ and hence that

(13)
$|e^{-\frac{\pi}{2}|\alpha-\ell_{0}/q|^{2}/(\varepsilon+\varepsilon^{-1}|\tau|^{2})}|\ll
e^{-c(2^{j-k})^{2}/2^{j-2k}}\ll e^{-c2^{j}}$

since $\varepsilon+\varepsilon^{-1}|\tau|^{2}\leq 2\cdot 2^{j}\varepsilon\ll
2^{j-2k}$. To estimate the sum where $\ell\neq\ell_{0}$ in (12) above we again
use the fact that $\varepsilon+\varepsilon^{-1}|\tau|^{2}\ll 2^{j-2k}$. Since
$|q\alpha-\ell|\geq 1/2$ for $\ell\neq\ell_{0}$ and $q|q_{j}$ with
$j\leq\log_{2}k-2$ it follows that

$q^{2}(\varepsilon+\varepsilon^{-1}|\tau|^{2})\ll(2^{2^{j}})^{2}2^{j-2k}\leq
2^{k}2^{-2k}\leq 2^{-k}$

and hence that

(14)
$\sum_{\ell\neq\ell_{0}}e^{-\frac{\pi}{2}|\alpha-\ell/q|^{2}/(\varepsilon+\varepsilon^{-1}|\tau|^{2})}\ll
2^{-c2^{k}}.$

Combining estimates (13), and (14) it follows that

$\|\widehat{s_{t}}\,1_{\Omega^{c}_{j,k}}\|_{\infty}\ll
q^{-d/2}(\varepsilon+|\tau|)^{-d/2}\,(e^{-c2^{j}}+2^{-c2^{k}})$

whenever $|\tau|\leq 2^{j/2}\varepsilon$ and $q|q_{j}$. Further combining this
with (11) we obtain that

$\displaystyle\sum_{q|q_{j}}\sum_{(a,q)=1}\int_{V_{a,q}}\|\widehat{s_{t}}\,1_{\Omega^{c}_{j,k}}\|_{\infty}dt$
$\displaystyle\ll\sum_{q|q_{j}}q^{-d/2+1}\varepsilon^{-(d-2)/2}\left[(e^{-c2^{j}}+2^{-c2^{k}})+2^{-j(d-2)/4}\right]$
$\displaystyle\ll\varepsilon^{-(d-2)/2}\,2^{-j(d-2)/4}\,\sum_{q|q_{j}}q^{-d/2+1}$
$\displaystyle\ll\varepsilon^{-(d-2)/2}\,2^{-j(d-2)/4}$

as the sum over $q$ converges for $d\geq 5.$

In order to estimate the second double sum in (10) we need the following
observation, whose proof we delay until after completing the proof of
Proposition 1.

######  Lemma 2.

Given $r>1$, $k\in\mathbb{N}$, then

$\sum_{q\nmid Q_{k}}q^{-r}\ll_{r}k^{-r+1}(\log k)^{-1}$

where $Q_{k}=\mathrm{lcm}\\{1\leq q\leq k\\}$ and $\ll_{r}$ denotes less than
a constant depending on $r$.

Using estimate (9) and Lemma 2 one can bound the second double sum in (10)
above by

$\sum_{q\nmid
q_{j}}\sum_{(a,q)=1}\int_{V_{a,q}}q^{-d/2}(\varepsilon+|\tau|)^{-d/2}dt\ll\varepsilon^{-(d-2)/2}\sum_{q\nmid
q_{j}}q^{-d/2+1}\ll\varepsilon^{-(d-2)/2}2^{-j(d-4)/2}j^{-1}$

whenever $d\geq 5$ completing the proof of Proposition 1.∎

###### Proof of Lemma 2.

If $q\in\mathbb{N}$ such that $q\nmid Q_{k}$, then either a large power of a
small prime divides $q$, or a large prime divides $q$. Explicitly, for prime
$p\leq k$, let $a_{p}=\min\\{a\in\mathbb{N}~{}:~{}k<p^{a}\\}.$ If $q\nmid
Q_{k}$ then one of the following must occur:

  1. (i)

there exists a $p>k$ such that $q=pq_{1}$

  2. (ii)

there exists a $p<k$ such that $q=p^{a_{p}}q_{1}$.

In the first case

$\displaystyle\sum_{\text{(i) holds}}q^{-r}$
$\displaystyle\leq\sum_{p>k}\sum_{q_{1}\in\mathbb{N}}p^{-r}q_{1}^{-r}$
$\displaystyle\ll_{r}\sum_{p>k}p^{-r}$ $\displaystyle\ll_{r}\sum_{m\geq
0}\sum_{p\in[2^{m}k,2^{m+1}k)}2^{-mr}k^{-r}$ $\displaystyle\ll_{r}\sum_{m\geq
0}2^{-mr}k^{-r}\frac{2^{m}k}{m\log(k)}\ll_{r}k^{-r+1}(\log k)^{-1}.$

In the second case

$\sum_{\text{(ii) holds}}q^{-r}\leq\sum_{p\leq
k}p^{-ra_{p}}\sum_{q_{1}\in\mathbb{N}}q_{1}^{-r}\ll_{r}\frac{k}{\log
k}k^{-r}.\qed$

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