#  Projected images of the Sierpinski tetrahedron and other fractal imaginary
cubes

Hideki Tsuiki
<EMAIL_ADDRESS>
###### Abstract

A projected image of a Sierpinski tetrahedron has a positive measure if and
only if the images $\mathrm{O,P,Q,R}$ of the four vertices satisfy
$p\vec{\mathrm{OP}}+q\vec{\mathrm{OQ}}+r\vec{\mathrm{OR}}=0$ for odd integers
$p,q,r$. This fact was essentially obtained by Kenyon (Symbolic dynamics and
its applications 135, 1992). We reformulate his proof through the notion of
projection of differenced radix expansion sets and obtain similar results for
other fractal objects. Specifically, we study projections of H-fractal and
T-fractal as well as Sierpinski tetrahedron, which are fractal imaginary cubes
(i.e., fractal objects that are projected to squares along three orthogonal
directions just as a cube) of the first two degrees, and characterize the
directions from which these fractals are projected to sets with positive
measures.

##  1 Introduction

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(a) | (b)

Figure 1: Projected images of the Sierpinski tetrahedron

A Sierpinski tetrahedron (i.e., three-dimensional Sierpinski gasket) is a
three-dimensional fractal object defined as the attractor of the iterated
function system (IFS) $\\{f_{d}(x)=\frac{x+d}{2}\mid d\in D_{\mathrm{S}}\\}$
where $D_{\mathrm{S}}\subset{\mathbb{R}}^{3}$ is the set of four vertices of a
regular tetrahedron (see Fig. 5(a)). That is, it is the unique non-empty
compact set $\mathrm{S}_{\infty}$ satisfying
$\mathrm{S}_{\infty}=\cup_{d\in{D_{\mathrm{S}}}}f_{d}(\mathrm{S}_{\infty})$
that exists due to the theory of self-similar fractals by Hutchinson [5].

A Sierpinski tetrahedron has a remarkable property that it is projected to
squares along the three orthogonal directions connecting the middle points of
opposite edges (Fig. 1(a)). An object that is projected to squares along three
orthogonal directions just as a cube is called an imaginary cube [15, 16] (See
Fig. 3 and 3 for examples of imaginary cubes and Definition 1 for the precise
definition of an imaginary cube), and Sierpinski tetrahedron is a fractal
imaginary cube. Moreover, as Fig. 1(b) shows, a Sierpinski tetrahedron is
projected to figures with positive Lebesgue measures along many other
directions, and they always form shapes that tile ${\mathbb{R}}^{2}$. In this
paper, we characterize directions from which Sierpinski tetrahedron and other
similar fractal objects are projected to sets with positive measures.

Because the IFS of a Sierpinski tetrahedron does not perform rotations,
projected images of a Sierpinski tetrahedron are two-dimensional fractals
generated by the same kind of IFS. For a set of points
$D\subset{\mathbb{R}}^{n}$ with cardinality $k^{2}$, let ${\cal F}^{n}(k,D)$
denote the fractal generated by an IFS of the form
$\\{f_{d}(x)=\frac{x+d}{k}\mid d\in{D}\\}$. A Sierpinski tetrahedron is ${\cal
F}^{3}(2,D_{\mathrm{S}})$ and its projected image by a projection $\varphi$ is
${\cal F}^{2}(2,\varphi(D_{\mathrm{S}}))$. Positively-measured fractals of the
form ${\cal F}^{2}(k,D)$ are special cases of self-affine tiles which have
been intensively studied from the 1990’s [1, 2, 6, 18, 19, 9, 11], and Kenyon
proved in [6] that for a four point set ${D}=\\{\mathrm{O,P,Q,R}\\}$, the
fractal ${\cal F}^{2}(2,D)$ has positive Lebesgue measure if and only if
$p\vec{\mathrm{OP}}+q\vec{\mathrm{OQ}}+r\vec{\mathrm{OR}}=0$ for odd integers
$p,q,r$. Since projections of a Sierpinski tetrahedron have this form, his
result characterizes the directions from which a Sierpinski tetrahedron is
projected to sets with positive measures.

We reformulate Kenyon’s proof through the notion of projections of differenced
radix expansion sets, and obtain similar results for other fractal imaginary
cubes. We call a fractal imaginary cube expressible as ${\cal F}^{3}(k,D)$ for
a digit set $D$ of cardinality $k^{2}$ a (dilational) fractal imaginary cube
of degree $k$. As we will explain in the next section, Sierpinski tetrahedron
is the only fractal imaginary cubes of degree 2. There are two fractal
imaginary cube of degree 3, which are called H-fractal and T-fractal([14, 16],
see Fig. 5(b, c)). We characterize the directions from which these two fractal
imaginary cubes are projected to sets with positive measures.

We introduce imaginary cubes and fractal imaginary cubes in the next section
and explain our results in Section 3. In section 4, we overview the properties
of self-affine tiles we use and give the key idea of projection of differenced
radix expansion sets. Then, we give proofs in Section 5. By a projection, we
mean not only an orthogonal projection but a projection to any plane that is
not parallel to the direction of the projection. This does not introduce
ambiguity to our results because the positivity of Lebesgue measure of a
projected image is independent of the choice of the plane.

##  2 Fractal Imaginary Cubes

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Regular tetrahedron | H | T

Figure 2: Examples of Imaginary cubes.

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Figure 3: Objects in Fig. 3 located in unit cubes.

Fractal objects with square projections along three orthogonal directions were
investigated in [14], and the more general notion of an imaginary cube was
introduced in [15].

######  Definition 1.

An imaginary cube is an object that is projected to squares along three
orthogonal directions so that, for each direction, the four edges of the
projected square are parallel to the other two directions.

An imaginary cube is contained in a cube defined by the three square
projections. When we need to specify this cube $C$, we say that it is an
imaginary cube of $C$. Note that a regular octahedron also has square
projections along three orthogonal directions, but it is not an imaginary
cube.

A regular tetrahedron is an example of a polyhedral imaginary cube. There are
two more polyhedral imaginary cubes that are relevant to this paper: hexagonal
bipyramid imaginary cube (H in short) and triangular antiprismoid imaginary
cube (T in short), which are defined in Fig. 3, and cubes of which they are
imaginary cubes are given in Fig. 3. H and T have some remarkable properties.
H is a double imaginary cube in that it is an imaginary cube of two different
cubes and thus it has six square projections. T has the property that the
three diagonals of T are orthogonal to each other, and H and T form a three-
dimensional tiling of the space [15].

The three imaginary cubes: regular tetrahedron, H and T, have the common
property that they are convex hulls of fractal imaginary cubes. According to
Hutchinson [5], for contractions $f_{i}:\mathbb{R}^{n}\to\mathbb{R}^{n}$
$(i=1,\dotsc,m)$, an IFS (iterated function system) $\\{f_{i}\mid
i=1,2,\ldots,m\\}$ defines a self-similar fractal object as the fixed point of
the following contraction map on the metric space of non-empty compact subsets
of $\mathbb{R}^{n}$.

$\displaystyle F(X)$ $\displaystyle=\bigcup_{i=1}^{m}f_{i}(X).$

In this article, we only consider the simple case $n$ is 2 or 3 and an IFS has
the form $\\{f_{d}(x)=\frac{x+d}{k}\mid d\in{D}\\}$ for some integer $k>1$ and
some ${D}\subset{\mathbb{R}}^{3}$ with $\lvert D\rvert=k^{2}$ called the digit
set. That is, the components of an IFS are dilations with ratio $\frac{1}{k}$
and center $\frac{d}{k-1}$. In this case, $F(X)$ can be expressed as

$\displaystyle F(X)$ $\displaystyle=\frac{X+D}{k}$ (1)

for $+$ the Minkowski sum. We denote the induced fractal object by ${\cal
F}(k,{D})$, or by ${\cal F}^{n}(k,D)$ when it is appropriate to make the
dimension of the space clear.

Fractals of the form $F(k,{D})$ have the following simple description. Let $A$
be any convex set that contains centers of $f_{d}$ for $d\in{D}$. We define a
sequence $(A_{i})_{0\leq i}$ of compact sets as $A_{0}=A$ and
$A_{i+1}=F(A_{i})$. We have $f_{d}(A)\subseteq A$ by the fact that $A$ is a
convex set that contains the fixed point of $f_{d}$. Therefore
$A_{1}=F(A_{0})=\cup_{d\in{D}}{f_{d}(A)}\subseteq A$. Thus, $F$ is monotonic
and we have a decreasing sequence $A_{0}\supseteq A_{1}\supseteq
A_{2}\supseteq\ldots$ of compact sets. Their intersection $\cap_{i\geq
0}A_{i}$ is a non-empty compact set that is a fixed point of $F$ (because any
point $x^{\prime}$ in $F(\cap_{i\geq 0}A_{i})$ has the form $\frac{x+d}{k}$
for $x\in\cap_{i\geq 0}A_{i}$, and thus $x^{\prime}\in\cap_{i\geq
0}\frac{A_{i}+D}{k}=\cap_{i\geq 0}A_{i+1}$). Thus, we have the description
${\cal F}(k,{D})=\cap_{i\geq 0}A_{i}$. As the starting object $A$, it is
natural to consider the polyhedron obtained as the convex hull of
$\\{\frac{d}{k-1}\mid d\in{D}\\}$, which is also the convex hull of the
fractal ${\cal F}(k,{D})$.

Note that ${\cal F}(k,{D})$ admits $k$-adic expansion $(d_{i})_{i>0}$ for
$d_{i}\in D$ as follows:

${\cal
F}(k,{D})=\Bigl{\\{}\sum_{i>0}\frac{d_{i}}{k^{i}}\Bigm{|}d_{i}\in{D}\Bigr{\\}}\,.$

Since our IFS consists of similarity maps that do not perform rotation, an
image of a three dimensional fractal ${\cal F}^{3}(k,D)$ by a projection
$\varphi$ is a two-dimensional fractal ${\cal F}^{2}(k,\varphi(D))$. In order
that ${\cal F}^{2}(k,\varphi(D))$ has a positive Lebesgue measure, its
Hausdorff dimension should be 2. On the other hand, the Hausdorff dimension of
${\cal F}^{2}(k,\varphi(D))$ is no more than $\log_{k}\lvert D\rvert$. Since
we are interested in fractals with minimum number of IFS-components, we only
study the case $\log_{k}\lvert D\rvert=2$, i.e., $\lvert D\rvert=k^{2}$. We
call an imaginary cube of the form ${\cal F}(k,{D})$ for $\lvert
D\rvert=k^{2}$ a dilational fractal imaginary cube of degree $k$, or simply a
fractal imaginary cube of degree $k$.

We study a condition on the digit set $D$ so that ${\cal F}^{3}(k,{D})$
becomes a fractal imaginary cube. If ${\cal F}^{3}(k,{D})$ is an imaginary
cube, then there is a cube $C$ of which ${\cal F}^{3}(k,{D})$ is an imaginary
cube. Since $C$ is a convex set containing ${\cal F}^{3}(k,{D})$, we can apply
the above construction for $A=C$. Therefore, for the sequence $C=C_{0}\supset
C_{1}\supset C_{2}\ldots$ defined as $C_{i+1}=F(C_{i})$, we have ${\cal
F}^{3}(k,{D})=\cap_{i\geq 0}\,C_{i}$. Since ${\cal F}^{3}(k,{D})$ and $C$ have
the same three square projections, $C_{i}$ are all imaginary cubes of $C$. In
particular, $C_{1}=F(C)$ is an imaginary cube of $C$. In the following lemma,
we show that it is also a sufficient condition.

######  Lemma 2.

If a digit set ${D}$ and a cube $C$ have the property that $F(C)$ is an
imaginary cube of $C$ for $F$ in (1), then ${\cal F}^{3}(k,{D})$ is a fractal
imaginary cube of $C$.

###### Proof.

We first show by induction that $C=C_{0}\supset C_{1}\supset\ldots$ for
$C_{i+1}=F(C_{i})$ are all imaginary cubes. Suppose that $C_{i}$ is an
imaginary cube of $C$ and $\varphi$ is a projection along an edge of $C$. Then
$C_{i}$ and $C$ are projected to the same square by $\varphi$. Thus,
$f_{d}(C_{i})$ and $f_{d}(C)$ are projected to the same square by $\varphi$.
Therefore, $C_{i+1}=\cup_{d\in D}f_{d}(C_{i})$ and $C_{1}=\cup_{d\in
D}f_{d}(C)$ are projected to the same figure by $\varphi$, which is a square
because $C_{1}$ is an imaginary cube of $C$. Thus, $C_{i+1}$ is an imaginary
cube. Now, we show that ${\cal F}^{3}(k,{D})=\cap_{i\geq 0}\,C_{i}$ is also an
imaginary cube. Let $B=\varphi(C)$ be the projected square. For every point
$p\in B$ and $i\geq 0$, $\varphi^{-1}(p)\cap C_{i}$ is a non-empty compact set
and $\varphi^{-1}(p)\cap C_{i}\supseteq\varphi^{-1}(p)\cap C_{i+1}$.
Therefore, $\varphi^{-1}(p)\cap(\cap_{i\geq 0}C_{i})=\cap_{i\geq
0}(\varphi^{-1}(p)\cap C_{i})$ is non-empty. Thus, $\varphi$ projects ${\cal
F}^{3}(k,{D})$ onto $B$. ∎

If ${D}$ a digit set of cardinality $k^{2}$ and $F(C)=\cup_{d\in D}f_{d}(C)$
is an imaginary cube, then $F(C)$ is a union of $k^{2}$ cubes selected from
the $k^{3}$ cubes obtained by cutting $C$ into $k\times k\times k$ small cubes
so that the $k^{2}$ cubes do not overlap when viewed from the three face-
directions of $C$. Such a selection exists corresponding to a Latin square of
size $k$, which is a $k\times k$ matrix of $K=\\{0,1,2,\ldots,k-1\\}$ such
that all the numbers appear on all the columns and rows. Thus, we have the
following theorem, where a Latin square is expressed as a function $h:K\times
K\to K$.

######  Theorem 3 ([14]).

Let $D\subset{\mathbb{R}}^{3}$ be a set with cardinality $k^{2}$. ${\cal
F}^{3}(k,{D})$ is an imaginary cube of the unit cube $[0,1]^{3}$ if and only
if ${D}=\\{(i,j,h(i,j))\mid 0\leq i,j\leq k-1\\}$ for a Latin square $h$.

When $k=2$, there are two digit sets of this form but they generate congruent
arrangements of the four cubes $F(C)$ and therefore there is only one fractal
imaginary cube of degree 2. We define the digit set

${D}_{\mathrm{S}}=\Bigl{\\{}\left(\\!\\!\begin{array}[]{r}1\\\ 0\\\
0\end{array}\\!\\!\right),\left(\\!\\!\begin{array}[]{r}0\\\ 1\\\
0\end{array}\\!\\!\right),\left(\\!\\!\begin{array}[]{r}0\\\ 0\\\
1\end{array}\\!\\!\right),\left(\\!\\!\begin{array}[]{r}1\\\ 1\\\
1\end{array}\\!\\!\right)\Bigr{\\}}.$

Fig. 5(a) shows $F(C)=\cup_{d\in D_{\mathrm{S}}}f_{d}(C)$ with the set
${D}_{\mathrm{S}}$ marked as red circles. ${D}_{\mathrm{S}}$ forms the set of
vertices of the regular tetrahedron $S$ in Fig. 5(a) and ${\cal
F}(2,{D}_{\mathrm{S}})$ is a Sierpinski tetrahedron.

(a) (b) (c)

Figure 4:  The arrangements of cubes by (a): ${D}_{\mathrm{S}}$,
(b):${D}_{\mathrm{H}}$, and (c):${D}_{\mathrm{T}}$.

(a) |
---|---
(b) |
(c) |

Figure 5:  The first two approximations of (a) $\mathrm{S}_{\infty}$, (b)
$\mathrm{H}_{\infty}$, and (c) $\mathrm{T}_{\infty}$.

When $k=3$, there are only two arrangements of such nine cubes modulo
congruence, and therefore there are two fractal imaginary cubes of degree 3.
This time, we consider the unit cube $[-1/2,1/2]^{3}$ centered at the origin
and consider Latin squares of $K=\\{-1,0,1\\}$ for the ease of developments in
the following sections. We define the following two digit sets
$D_{\mathrm{H}}$ and $D_{\mathrm{T}}$ in $\\{-1,0,1\\}^{3}$:

$\displaystyle\begin{split}{D}_{\mathrm{H}}=&\Bigl{\\{}\left(\\!\\!\begin{array}[]{r}0\\\
0\\\ 0\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}0\\\ \pm 1\\\
\mp 1\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}\pm 1\\\ 0\\\
\mp 1\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}\pm 1\\\ \mp
1\\\ 0\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}\pm 1\\\ \pm
1\\\ \pm 1\end{array}\\!\\!\right)\Bigr{\\}}\ \ \text{(double signs of the
same orders)}\\\
{D}_{\mathrm{T}}=&\Bigl{\\{}\left(\\!\\!\begin{array}[]{r}1\\\ -1\\\
-1\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}-1\\\ 1\\\
-1\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}-1\\\ -1\\\
1\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}-1\\\ 0\\\
0\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}0\\\ -1\\\
0\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}0\\\ 0\\\
-1\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}0\\\ 1\\\
1\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}1\\\ 0\\\
1\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}1\\\ 1\\\
0\end{array}\\!\\!\right)\Bigr{\\}}.\ \end{split}$

The corresponding sets $\cup_{d\in D_{\mathrm{H}}}f_{d}(C)$ and $\cup_{d\in
D_{\mathrm{T}}}f_{d}(C)$ are depicted in Fig. 5(b, c). Here, the fixed points
of $f_{d}$ for $d\in{D}_{\mathrm{H}}$ and $f_{d}$ for $d\in{D}_{\mathrm{T}}$
are indicated as red circles, and they are the sets
$\frac{{D}_{\mathrm{H}}}{2}$ and $\frac{{D}_{\mathrm{T}}}{2}$ respectively.
$\frac{{D}_{\mathrm{H}}}{2}$ is the set of eight vertices and the center of
the dodecahedron H in Fig. 5(b). $\frac{{D}_{\mathrm{T}}}{2}$ is the set of
six vertices and the three middle points of the edges of the larger regular
triangle face of the octahedron T in Fig. 5(c). ${\cal F}(3,{D}_{\mathrm{H}})$
and ${\cal F}(3,{D}_{\mathrm{T}})$ are fractal imaginary cubes whose convex
hulls are H and T, respectively. We denote ${\cal F}(2,{D}_{\mathrm{S}})$,
${\cal F}(3,{D}_{\mathrm{H}})$ and ${\cal F}(3,{D}_{\mathrm{T}})$ by
$\mathrm{S}_{\infty}$, $\mathrm{H}_{\infty}$ and $\mathrm{T}_{\infty}$,
respectively. Fig. 5 shows the first two levels of the approximations of these
three fractals starting with their convex hulls. We call the shapes (i.e.,
equivalence class module congruence and similarity) of ${\cal
F}(3,{D}_{\mathrm{H}})$ and ${\cal F}(3,{D}_{\mathrm{T}})$ H-fractal and
T-fractal, respectively. Note that an H-fractal has six-fold symmetry and it
is a double imaginary cube.

##  3 Main results

As imaginary cubes, the fractals $\mathrm{S}_{\infty}$, $\mathrm{H}_{\infty}$,
and $\mathrm{T}_{\infty}$ have orthogonal projections to squares along the
vectors $(1,0,0),(0,1,0)$ and $(0,0,1)$. In addition, since
$\mathrm{H}_{\infty}$ is a double imaginary cube, it is projected to squares
along $(-1,2,2),(2,-1,2)$ and $(2,2,-1)$ which are obtained by rotating the
above directions by 180 (or equivalently 60) degrees around the axis
$(1,1,1)$.

In addition, these fractals are projected to figures with positive measures in
many other directions. For example, $\mathrm{S}_{\infty}$ is projected to Fig.
6(a) from the vertices of the cube, and $\mathrm{H}_{\infty}$ and
$\mathrm{T}_{\infty}$ are projected to (b) and (c), respectively, along
$(-1,1,1)$, $(1,-1,1)$ and $(1,1,-1)$. Also, $\mathrm{H}_{\infty}$ and
$\mathrm{T}_{\infty}$ are projected to (e) and (f), respectively, along
$(0,1,1)$, $(1,0,1)$ and $(1,1,0)$. $\mathrm{S}_{\infty}$ is projected to (d)
along $(2,1,0)$ and its permutations. In addition, $\mathrm{H}_{\infty}$ is
projected to

(b) along $(1,1,-5),(1,-5,1)$ and $(-5,1,1)$ and (e) along $(4,1,1),(1,4,1)$
and $(1,1,4)$ which are obtained by rotating the above directions by 180
degree.

Our main theorem is the characterization of the directions along which
Sierpinski tetrahedron, H-fractal, T-fractal are projected to figures with
positive Lebesgue measures:

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(a) | (b) | (c)
|  |
(d) | (e) | (f)
Figure 6:  Projected images of (a) $\mathrm{S}_{\infty}$ from $(1,-1,1)$; (b)
$\mathrm{H}_{\infty}$ from $(1,-1,1)$;
(c) $\mathrm{T}_{\infty}$ from $(1,-1,1)$; (d) $\mathrm{S}_{\infty}$ from
$(1,2,0)$; (e) $\mathrm{H}_{\infty}$ from $(1,1,0)$;
(f) $\mathrm{T}_{\infty}$ from $(1,1,0)$.

######  Theorem 4.

(1) Let $a,b,c$ be coprime integers.

>   * (a)
>
> $\mathrm{S}_{\infty}$ is projected along the vector $(a,b,c)$ to a set with
> positive Lebesgue measure if and only if $a+b+c$ is an odd number.
>
>   * (b)
>
> $\mathrm{T}_{\infty}$ is projected along the vector $(a,b,c)$ to a figure
> with positive Lebesgue measure if and only if $3\nmid a+b+c$.
>
>   * (c)
>
> $\mathrm{H}_{\infty}$ is projected along the vector $(a,b,c)$ to a figure
> with positive Lebesgue measure if and only if $3\nmid a+b+c$ or $a\equiv
> b\equiv c\equiv-\frac{a+b+c}{3}\not\equiv 0\mod 3$.
>
>

(2) For real numbers $a,b,c$ such that $a/b$, $b/c$, or $c/a$ exist and are
irrational, $\mathrm{S}_{\infty}$, $\mathrm{T}_{\infty}$, and
$\mathrm{H}_{\infty}$ are projected along $(a,b,c)$ to null-sets.

As Theorem 4(1c) says, the directions along which $\mathrm{H}_{\infty}$ is
projected to sets with positive measures are divided into two disjoint sets.
The first one is the same as that of $\mathrm{T}_{\infty}$ and the second one
is obtained by rotating the first one by 180 degree around the axis $(1,1,1)$.
We first show this fact.

######  Lemma 5.

An integer vector $(a,b,c)$ such that $3\nmid a+b+c$ is rotated by a 180-degee
rotation around the axis $(1,1,1)$ to $\frac{1}{3}(\alpha,\beta,\gamma)$ for
integers $\alpha,\beta,\gamma$ such that
$\alpha\equiv\beta\equiv\gamma\equiv-\frac{\alpha+\beta+\gamma}{3}\not\equiv
0\mod 3$, and vice versa.

###### Proof.

Since $\frac{1}{3}(-1,2,2)$, $\frac{1}{3}(2,-1,2)$ and $\frac{1}{3}(2,2,-1)$
are obtained by rotating $(1,0,0)$, $(0,1,0)$ and $(0,0,1)$ respectively, we
set

$(\alpha,\beta,\gamma)=a(-1,2,2)+b(2,-1,2)+c(2,2,-1)$

for $a,b,c\in{\mathbb{R}}$ and show that $a,b,c$ are integers and $3\nmid
a+b+c$ if and only if $\alpha,\beta,\gamma$ are integers and
$\alpha\equiv\beta\equiv\gamma\equiv-\frac{\alpha+\beta+\gamma}{3}\not\equiv
0\mod 3$.

If $a+b+c\equiv 1\mod 3$, we have $\alpha+\beta+\gamma=3(a+b+c)\equiv 3\mod 9$
and thus $\frac{\alpha+\beta+\gamma}{3}\equiv 1\mod 3$. Since
$\alpha=2(a+b+c)-3a$, we have $\alpha\equiv-1\mod 3$. In the same way,
$\alpha\equiv\beta\equiv\gamma\equiv-1\mod 3$.

Similarly, if $a+b+c\equiv-1\mod 3$, we have $\alpha+\beta+\gamma\equiv-3\mod
9$ and therefore $\frac{\alpha+\beta+\gamma}{3}\equiv-1\mod 3$ and
$\alpha\equiv\beta\equiv\gamma\equiv 1\mod 3$.

On the other hand, if
$\alpha\equiv\beta\equiv\gamma\equiv-\frac{\alpha+\beta+\gamma}{3}\not\equiv
0\mod 3$, then $a+b+c=\frac{\alpha+\beta+\gamma}{3}\not\equiv 0\mod 3$.
$a=\frac{1}{3}(2\frac{\alpha+\beta+\gamma}{3}-\alpha)$ is an integer, and
similarly for $b$ and $c$. ∎

If the direction of the projection is parallel to the plane $x+y+z=0$, then
the projected images of $\mathrm{S}_{\infty}$, $\mathrm{H}_{\infty}$, and
$\mathrm{T}_{\infty}$ are obviously null-sets and the direction $(a,b,c)$ of
the projection satisfies $a+b+c=0$. Therefore, Theorem 4 holds for this case.

If the direction is not parallel to $x+y+z=0$, then one can restate Theorem 6
as a condition on $a,b$ and $c$ so that the projected image to the plane
$x+y+z=0$, that has the form ${\cal F}^{2}(k,D^{\prime})$ for some two-
dimensional digit set $D^{\prime}$ parameterized by $a,b$ and $c$, has
positive measure. Note that for a non-degenerate affine transformation $\psi$
and a two-dimensional digit set $D^{\prime}$ we have ${\cal
F}(k,\psi(D^{\prime}))=\psi({\cal F}(k,D^{\prime}))$, and thus ${\cal
F}(k,D^{\prime})$ has positive measure if and only if ${\cal
F}(k,\psi(D^{\prime}))$ does. Therefore, we can further restate it as a
property of an affine-transformed image of the projected image. In addition,
we can reduce the number of parameters to two and use two real parameters $u$
and $v$ to express the shape of the projected image because we have solved the
case $a+b+c=0$.

With this idea, we define the following three series of digit sets in
${\mathbb{R}}^{2}$.

$\displaystyle\begin{split}D^{u,v}_{\mathrm{S}}&=\Bigl{\\{}\left(\\!\\!\begin{array}[]{c}1\\\
0\\\ \end{array}\\!\\!\right),\left(\\!\\!\begin{array}[]{c}0\\\ 1\\\
\end{array}\\!\\!\right),\left(\\!\\!\begin{array}[]{c}0\\\ 0\\\
\end{array}\\!\\!\right),\left(\\!\\!\begin{array}[]{c}u\\\ v\\\
\end{array}\\!\\!\right)\Bigr{\\}}\\\
D^{u,v}_{\mathrm{H}}&=\Bigl{\\{}\left(\\!\\!\begin{array}[]{c}0\\\ 0\\\
\end{array}\\!\\!\right),\left(\\!\\!\begin{array}[]{c}0\\\ \pm 1\\\
\end{array}\\!\\!\right),\left(\\!\\!\begin{array}[]{c}\pm 1\\\ 0\\\
\end{array}\\!\\!\right),\left(\\!\\!\begin{array}[]{c}\pm 1\\\ \mp 1\\\
\end{array}\\!\\!\right),\left(\\!\\!\begin{array}[]{c}\pm u\\\ \pm v\\\
\end{array}\\!\\!\right)\Bigr{\\}}\\\
D^{u,v}_{\mathrm{T}}&=\Bigl{\\{}\left(\\!\\!\begin{array}[]{c}1\\\ -1\\\
\end{array}\\!\\!\right),\left(\\!\\!\begin{array}[]{c}-1\\\ 1\\\
\end{array}\\!\\!\right),\left(\\!\\!\begin{array}[]{c}-1\\\ -1\\\
\end{array}\\!\\!\right),\left(\\!\\!\begin{array}[]{c}-1\\\ 0\\\
\end{array}\\!\\!\right),\left(\\!\\!\begin{array}[]{c}0\\\ -1\\\
\end{array}\\!\\!\right),\left(\\!\\!\begin{array}[]{c}0\\\ 0\\\
\end{array}\\!\\!\right),\\\ &\hskip
170.71652pt\left(\\!\\!\begin{array}[]{c}-1+u\\\ v\\\
\end{array}\\!\\!\right),\left(\\!\\!\begin{array}[]{c}u\\\ -1+v\\\
\end{array}\\!\\!\right),\left(\\!\\!\begin{array}[]{c}u\\\ v\\\
\end{array}\\!\\!\right)\Bigr{\\}}\end{split}$

For $D_{\mathrm{S}}$, we consider projections to the plane $x+y+z=1$. This
plane contains three points $(1,0,0),(0,1,0),(0,0,1)$ of $D_{\mathrm{S}}$,
which are fixed by projections. Therefore, we consider an (affine-transformed)
two-dimensional coordinate on $x+y+z=1$ that assigns $(0,0)$, $(1,0)$, $(0,1)$
to $(0,0,1),(1,0,0),(0,1,0)$, respectively. $D^{u,v}_{\mathrm{S}}$ is an
affine-transformed projected image of ${D}_{\mathrm{S}}$ by setting $(u,v)$ to
be the address of the projected image of $(1,1,1)$.

$D^{u,v}_{\mathrm{H}}$ is similarly defined from $D_{\mathrm{H}}$ for the
plane $x+y+z=0$. The plane $x+y+z=0$ contains seven points of
$D_{\mathrm{H}}$, which are fixed by projections, so we consider an (affine-
transformed) two-dimensional coordinate on $x+y+z=0$ that assigns $(0,0)$,
$(1,0)$, $(0,1)$ to $(0,0,0)$ $(-1,0,1)$, $(0,-1,1)$, respectively.
$D^{u,v}_{\mathrm{H}}$ is an affine-transformed projected image of
${D}_{\mathrm{H}}$ by setting $(u,v)$ to be the address of the projected image
of $(1,1,1)$.

For $D^{u,v}_{\mathrm{T}}$, the plane $x+y+z=-1$ contains six points of
$D_{\mathrm{T}}$ which are fixed by projections, so we consider an (affine-
transformed) two-dimensional coordinate on $x+y+z=-1$ that assigns $(0,0)$,
$(1,0)$, $(0,1)$ to $(0,0,-1)$, $(1,0,0),(0,1,0)$, respectively.
$D^{u,v}_{\mathrm{T}}$ is an affine-transformed projected image of
${D}_{\mathrm{T}}$ by setting $(u,v)$ to be the address of the projected image
of $(1,1,0)$.

######  Theorem 6.

  * (a)

${\cal F}(2,D^{u,v}_{\mathrm{S}})$ has a positive measure if and only if
$u=\frac{p}{r}$ and $v=\frac{q}{r}$ for odd coprime integers $p,q,r$.

  * (b)

${\cal F}(3,D^{u,v}_{\mathrm{T}})$ has a positive measure if and only if
$u=\frac{p}{r}$ and $v=\frac{q}{r}$ for coprime integers $p,q,r$ such that
$p\equiv q\equiv r\mod 3$.

  * (c)

${\cal F}(3,D^{u,v}_{\mathrm{H}})$ has a positive measure if and only if
$u=\frac{p}{r}$ and $v=\frac{q}{r}$ for coprime integers $p,q,r$ such that
$p\equiv q\equiv r\mod 3$ or $p\equiv q\equiv-r\mod 3$.

Theorem 6(a) was proved by Kenyon in [6, Theorem 14]. Note that the following
corollary holds since an arbitrary non-colinear four point set
$D\subset{\mathbb{R}}^{2}$ is affine transformed to $D^{u,v}_{\mathrm{S}}$.

######  Corollary 7.

Let ${D}=\\{\mathrm{O,P,Q,R}\\}\subset{\mathbb{R}}^{2}$ be a non-colinear
four-points set. ${\cal F}(2,{D})$ has a positive Lebesgue measure if and only
if $p\vec{\mathrm{OP}}+q\vec{\mathrm{OQ}}+r\vec{\mathrm{OR}}=0$ for odd
integers $p,q,r$.

######  Proposition 8.

Theorem 6 and Theorem 4 are equivalent.

###### Proof.

If $\varphi$ is parallel to the line $x+y+z=1$, that is, if $a+b+c=0$, then
Theorem 4 holds as we have noted. We show the equivalence for other cases.

Case $\mathrm{S}_{\infty}$: One can calculate

$u=\frac{-a+b+c}{a+b+c},\ v=\frac{a-b+c}{a+b+c}.$

In addition, for $p=-a+b+c$, $q=a-b+c$ and $r=a+b+c$, we have
$a=\frac{r-p}{2}$, $b=\frac{r-q}{2}$ and $c=\frac{p+q}{2}$. Therefore, one can
easily show that (1) $u=\frac{p}{r}$ or $v=\frac{q}{r}$ is irrational if and
only if $\frac{a}{b}$ or $\frac{b}{c}$, or $\frac{c}{a}$ exists and is
irrational; (2) $a,b,c$ are coprime integers such that $a+b+c$ is odd if and
only if $p,q,r$ are odd coprime integers.

Case $\mathrm{T}_{\infty}$: One can calculate

$u=\frac{-2a+b+c}{a+b+c},\ v=\frac{a-2b+c}{a+b+c}.$

In addition, for $p=-2a+b+c$, $q=a-2b+c$ and $r=a+b+c$, we have
$a=\frac{-p+r}{3}$, $b=\frac{-q+r}{3}$ and $c=\frac{p+q+r}{3}.$ From these
facts, we can prove the above (1) and the following (2’): $a,b,c$ are coprime
integers such that $3\nmid a+b+c$ if and only if $p,q,r$ are coprime integers
such that $p\equiv q\equiv r\mod 3$.

Case $\mathrm{H}_{\infty}$: We have the same $u$ and $v$ as
$\mathrm{T}_{\infty}$. Thus, we have (1) and (2’). In addition, there is
another direction along which ${\cal F}^{3}(3,D_{\mathrm{H}})$ is projected to
${\cal F}^{2}(3,D^{u,v}_{\mathrm{H}})$. That is, $(-1,-1,-1)$ is mapped to
$(u,v)$ and $(1,1,1)$ is mapped to $(-u,-v)$. In this case,

$u=\frac{2a-b-c}{a+b+c},\ v=\frac{-a+2b-c}{a+b+c}.$

Then, for $p=\frac{2a-b-c}{3}$, $q=\frac{-a+2b+c}{3}$ and $r=\frac{a+b+c}{3}$,
we have $a=r+p$, $b=r+q$, and $c=p+q+r$. In this case, one can show (2”):
$a,b,c$ are coprime integers such that $a\equiv b\equiv
c\equiv-\frac{a+b+c}{3}\mod 3$ if and only if $p,q,r$ are coprime integers
such that $p\equiv q\equiv-r\mod 3$. ∎

##  4 Differenced radix expansion sets

As is mentioned in the introduction, positively-measured fractals of the form
${\cal F}^{2}(k,D)$ are special cases of self-affine tiles. A self-affine tile
is a compact set $T$ in ${\mathbb{R}}^{2}$ of positive Lebesgue measure such
that $A(T)=\cup_{d\in D}(T+d)$ for an expansive matrix $A$ and a digit set
$D\subset{\mathbb{R}}^{2}$ of cardinality $\lvert\det(A)\rvert$, and ${\cal
F}^{2}(k,D)$ is a special case that $A=\left(\begin{array}[]{cc}k&0\\\
0&k\end{array}\right)$.

Characterization of pairs $(A,D)$ that generate self-affine tiles was studied
by Bandit [1], Kenyon [6], Lagarias and Wang [9], and others. In our study, we
use the following characterization in [9]. Here, we state it only for self-
affine tiles that have the form ${\cal F}^{2}(k,D)$.

######  Theorem 9 (Theorem 1.1 of [9]).

Let $k\geq 2$, ${D}\subset\mathbb{R}^{2}$ be a set of cardinality $k^{2}$ and
${\bf 0}\in{D}$. The following four conditions are equivalent.

  * (i)

${\cal F}^{2}(k,D)$ has positive Lebesgue measure.

  * (ii)

${\cal F}^{2}(k,D)$ has nonempty interior.

  * (iii)

${\cal F}^{2}(k,D)$ is the closure of its interior, and its boundary has
Lebesgue measure zero.

  * (iv)

For each $t\geq 1$, all the $(k,D)$-expansions of length $t$ designate
distinct points in ${\cal E}^{2}(k,D,t)$, and ${\cal E}^{2}(k,D)$ is a
uniformly discrete set, i.e., there exists $\delta>0$ such that $(\lvert
x-x^{\prime}\rvert>\delta\lor x=x^{\prime})$ for $\forall x,x^{\prime}\in{\cal
E}^{2}(k,D)$.

Here, $(k,D)$-expansion, ${\cal E}(k,D,t)\subset{\mathbb{R}}^{n}$ and ${\cal
E}(k,D)\subset{\mathbb{R}}^{n}$ are defined as follows. If
$x\in{\mathbb{R}}^{n}$ is expressed as $x=\sum_{j=0}^{t-1}k^{j}d_{j}$ for
$d_{j}\in D$, then we say that the sequence $(d_{j})_{0\leq j<t}$ is a
$(k,D)$-expansion of $x$. We define

$\displaystyle{\cal E}(k,D,0)$ $\displaystyle=\\{{\bf 0}\\}\,,$
$\displaystyle{\cal E}(k,D,t)$ $\displaystyle=k{\cal E}(k,D,t-1)+D\,\ (t>0)$

We have ${\cal E}(k,D,0)\subseteq{\cal E}(k,D,1)\subseteq\ldots$ and

$\displaystyle{\cal E}(k,D)$ $\displaystyle=\bigcup_{t=1}^{\infty}{{\cal
E}(k,D,t)}\,.$ (2)

We sometimes write ${\cal E}^{n}(k,D)$ for ${\cal E}(k,D)$ to make the
dimension of the space clear. We have

$\displaystyle{\cal E}(k,D,t)=\\{\sum_{j=0}^{t-1}k^{j}d_{j}\mid
d_{j}\in{D}\\}\,$

and thus ${\cal E}(k,D,t)$ is the set of points that have $(k,D)$-expansions
of length $t$.

######  Corollary 10 (Corollary 1.1 of [9]).

Let ${D}\subset{\mathbb{Z}}^{2}$ be a set of cardinality $k^{2}$ and ${\bf
0}\in D$. If ${D}$ forms a complete residue system of
${\mathbb{Z}}^{2}/(k{\mathbb{Z}}\times k{\mathbb{Z}})$, then ${\cal
F}^{2}(k,D)$ has a positive Lebesgue measure.

###### Proof.

We use Theorem 9 (iv) $\to$ (i). Suppose that ${D}$ forms a complete residue
system of ${\mathbb{Z}}^{2}/(k{\mathbb{Z}}\times k{\mathbb{Z}})$. ${\cal
E}^{2}(k,D)$ is a uniformly discrete set because $D$ is an integral digit set.
If two $(k,D)$-expansions $(d_{i})_{i<t}$ and $(e_{i})_{i<t}$ designate the
same point, i.e., $\sum_{i=0}^{t-1}k^{i}d_{i}=\sum_{i=0}^{t-1}k^{i}e_{i}$,
then $k(\sum_{i=1}^{t-1}k^{i-1}(d_{i}-e_{i}))+(d_{0}-e_{0})=0$ and therefore,
$d_{0}-e_{0}\in k{\mathbb{Z}}\times k{\mathbb{Z}}$ which implies
$d_{0}=e_{0}$. We can inductively show that $d_{i}=e_{i}$ for $0\leq i<t$. ∎

With this Corollary, one can derive the ’if’ part of Theorem 6.

######  Lemma 11.

The ’if’ parts of Theorem 6 (a), (b), (c) hold.

###### Proof.

For (a), consider the digit set

${D}^{\prime}=\Bigl{\\{}\left(\\!\\!\begin{array}[]{c}0\\\ 0\\\
\end{array}\\!\\!\right),\left(\\!\\!\begin{array}[]{c}r\\\ 0\\\
\end{array}\\!\\!\right),\left(\\!\\!\begin{array}[]{c}0\\\ r\\\
\end{array}\\!\\!\right),\left(\\!\\!\begin{array}[]{c}p\\\ q\\\
\end{array}\\!\\!\right)\Bigr{\\}}$

that generates $r{\cal F}(2,{D}^{u,v}_{\mathrm{S}})$ for $u=\frac{p}{r}$ and
$v=\frac{q}{r}$. If $p,q,r$ are odd, then $D^{\prime}$ is a complete residue
system of ${\mathbb{Z}}^{2}/(2{\mathbb{Z}}\times 2{\mathbb{Z}})$. Proofs of
(b) and (c) are similar. ∎

For an integral digit set $D\subset{\mathbb{Z}}^{n}$, a differenced digit set
$\Delta(D)$ is the digit set

$\Delta(D)=D-D=\\{x-y\mid x,y\in D\\}$

and differenced radix expansion set is the set ${\cal E}(k,\Delta(D))$. Note
that ${\bf 0}\in\Delta(D)$ even if ${\bf 0}\not\in D$. Differenced radix
expansion set is an important tool for investigating tiling properties of
self-affine tiles [6, 9].

Theorem 9 says that we can prove that ${\cal F}^{2}(k,D)$ is a null-set by
showing the existence of different $(k,D)$-expansions of the same point, that
is, the existence of different sequences $(d_{i})_{0\leq i<t}$ and
$(e_{i})_{0\leq i<t}$ of ${D}$ such that

$\sum_{i=0}^{t-1}k^{i}d_{i}=\sum_{i=0}^{t-1}k^{i}e_{i}\,,$

or equivalently,

$\sum_{i=0}^{t-1}k^{i}(d_{i}-e_{i})={\bf 0}.$

It means that ${\bf 0}$ has a $(k,\Delta(D))$-expansion of length $t$ other
than $({\bf 0})_{0\leq i<t}$. Thus, one can show that ${\cal F}^{2}(k,D)$ is a
null-set through the observation of differenced radix expansions of ${\bf 0}$,
and the proof of Theorem 6(a) in [6] by Kenyon was based on this idea. Here,
we reformulate his proof so that it uses projection of three-dimensional
differenced radix expansion sets.

For a three-dimensional digit set $D\subset{\mathbb{R}}^{3}$ and a projection
$\varphi$, we have

$\Delta(\varphi({D}))=\varphi(\Delta({D}))\,.$

In addition, we have

${\cal E}^{2}(k,\varphi({D}))=\varphi({\cal E}^{3}(k,{D}))\,.$

Therefore, we have

${\cal E}^{2}(k,\Delta(\varphi({D})))=\varphi({\cal E}^{3}(k,\Delta({D})))\,.$

Moreover, any differenced radix expansion of $(0,0)$ by $\varphi(D)$ is
obtained as a projection of a differenced radix expansion of
$\varphi^{-1}((0,0))$ by $D$. If $\varphi$ is a projection along $(a,b,c)$ for
coprime integers $a,b,c$, then $\varphi^{-1}((0,0))=\\{(ma,mb,mc)\mid
m\in{\mathbb{Z}}\\}$. Therefore, we have the following lemma. Note that ${\cal
E}^{2}(k,\varphi(D))$ is always a uniformly discrete set for an integral digit
set $D$ and a projection $\varphi$ along an integral vector.

######  Lemma 12.

Let $a,b,c$ be integers and $\varphi$ be a projection along $(a,b,c)$. ${\cal
F}^{3}(k,D)$ is projected to a null-set by $\varphi$ if and only if
$(ma,mb,mc)\in{\cal E}^{3}(k,\Delta(D))$ for some $m\neq 0$ or ${\bf
0}\in{\mathbb{R}}^{3}$ has a $(k,\Delta(D))$-expansion other than $({\bf
0})_{0\leq i<t}$.

With this lemma, for $D\in\\{D_{\mathrm{S}},D_{\mathrm{T}},D_{\mathrm{H}}\\}$,
one can study the directions along which ${\cal F}^{2}(k,D)$ is projected to a
null-set by analyzing the shape of the differenced radix expansion set ${\cal
E}^{3}(k,\Delta(D))$. Here, since these expansion sets are rather complicated,
we consider other digit sets
$D_{\mathrm{S}^{\prime}},D_{\mathrm{T}^{\prime}},D_{\mathrm{H}^{\prime}}$ that
are also projected to
$D^{u,v}_{\mathrm{S}},D^{u,v}_{\mathrm{T}},D^{u,v}_{\mathrm{H}}$,
respectively, and analyze ${\cal E}^{3}(k,\Delta(D))$ for
$D\in\\{D_{\mathrm{S}^{\prime}},D_{\mathrm{T}^{\prime}},D_{\mathrm{H}^{\prime}}\\}$.

##  5 Proofs of the main theorem

We study the ’only if’ parts of Theorem 6.

|
---|---
(a) $D_{\mathrm{S}^{\prime}}$ | (b) $\Delta(D_{\mathrm{S}^{\prime}})$
Figure 7:  (a) The digit set $D_{\mathrm{S}^{\prime}}$; (b) The differenced
digit set
$\Delta(D_{\mathrm{S}^{\prime}})={D}_{\mathrm{S}^{\prime}}+(-{D}_{\mathrm{S}^{\prime}})$.
Figure 8:  ${\cal E}(2,\Delta({D}_{\mathrm{S}^{\prime}}))$ for $-2\leq x,y\leq
6$, $-1\leq z\leq 4$. Points in $2{\cal
E}(2,\Delta({D}_{\mathrm{S}^{\prime}}))$ are circled and translations of
$\Delta({D}_{\mathrm{S}^{\prime}})$ by $(0,0,0)$ and $(0,4,2)$ are colored
red.

###  5.1 Sierpinski Tetrahedron case:

We consider the three-dimensional digit set

${D}_{\mathrm{S}^{\prime}}=\Bigl{\\{}\left(\\!\\!\begin{array}[]{r}0\\\ 0\\\
0\end{array}\\!\\!\right),\left(\\!\\!\begin{array}[]{r}1\\\ 0\\\
0\end{array}\\!\\!\right),\left(\\!\\!\begin{array}[]{r}0\\\ 1\\\
0\end{array}\\!\\!\right),\left(\\!\\!\begin{array}[]{r}0\\\ 0\\\
1\end{array}\\!\\!\right)\Bigr{\\}}$

which is projected on the $xy$-plane along $(a,b,c)$ to $D^{u,v}_{\mathrm{S}}$
for $u=-\frac{a}{c}$ and $v=-\frac{b}{c}$. The sets
${D}_{\mathrm{S}^{\prime}}$ and $\Delta({D}_{\mathrm{S}^{\prime}})$ are
depicted in Fig. 7. We define

$\displaystyle B$ $\displaystyle=\\{(2^{n}x,2^{n}y,2^{n}z)\mid\text{$x,y,z$
are odd}\\},$ $\displaystyle C$ $\displaystyle={\mathbb{Z}}^{3}\setminus B.$

######  Lemma 13.

${\cal E}(2,\Delta({D}_{\mathrm{S}^{\prime}}))=C$.

###### Proof.

${\cal E}(2,\Delta({D}_{\mathrm{S}^{\prime}}))\subseteq C$ is proved by
induction on ${\cal E}(2,\Delta({D}_{\mathrm{S}^{\prime}}))$. For the proof of
${\cal E}(2,\Delta({D}_{\mathrm{S}^{\prime}}))\supseteq C$, we prove

$(x,y,z)\in C\to(x,y,z)\in{\cal E}(2,\Delta({D}_{\mathrm{S}^{\prime}}))$

by well-founded induction on $\lvert x\rvert+\lvert y\rvert+\lvert z\rvert$.
It holds for the case $\lvert x\rvert+\lvert y\rvert+\lvert z\rvert=0$ because
$(0,0,0)\in C$ and $(0,0,0)\in{\cal E}(2,\Delta({D}_{\mathrm{S}^{\prime}}))$.
For $n>0$, suppose that it holds for the case $\lvert x\rvert+\lvert
y\rvert+\lvert z\rvert<n$ and show that it holds for the case $\lvert
x\rvert+\lvert y\rvert+\lvert z\rvert=n$. Suppose that $(x,y,z)\in C$ and
$\lvert x\rvert+\lvert y\rvert+\lvert z\rvert=n>0$. We define
$(x^{\prime},y^{\prime},z^{\prime})\in C$ such that $\lvert
x^{\prime}\rvert+\lvert y^{\prime}\rvert+\lvert z^{\prime}\rvert<n$ as
follows.

If $x$, $y$ and $z$ are all even, then
$(x^{\prime},y^{\prime},z^{\prime})=(x/2,y/2,z/2)$. If $x,y$ and $z$ are all
odd, then $(x,y,z)\not\in C$. For the other cases, without loss of generality,
we consider the following two cases.

Case $x$ and $y$ are even and $z$ is odd: If
$(x,y,z)\in\\{(0,0,1),(0,0,-1)\\}$, then
$(x^{\prime},y^{\prime},z^{\prime})=(0,0,0)$. Otherwise, at least one of
$(x/2,y/2,(z-1)/2)$ or $(x/2,y/2,(z+1)/2)$ is in $C$ and let
$(x^{\prime},y^{\prime},z^{\prime})$ be such an element.

Case $x$ is even and $y$ and $z$ are odd: If
$(x,y,z)\in\\{(0,1,-1),(0,-1,1)\\}$, then
$(x^{\prime},y^{\prime},z^{\prime})=(0,0,0)$. Otherwise, at least one of
$(x/2,(y-1)/2,(z+1)/2)$ and $(x/2,(y+1)/2,(z-1)/2)$ is in $C$, and let
$(x^{\prime},y^{\prime},z^{\prime})$ be such an element.

By induction hypothesis, $(x^{\prime},y^{\prime},z^{\prime})\in{\cal
E}(2,\Delta({D}_{\mathrm{S}^{\prime}}))$. Since
$(x,y,z)-2(x^{\prime},y^{\prime},z^{\prime})\in\Delta({D}_{\mathrm{S}^{\prime}})$
for every case, $(x,y,z)\in{\cal E}(2,\Delta({D}_{\mathrm{S}^{\prime}}))$. ∎

Now, we prove the ’only if’ part of Theorem 6(a):

######  Lemma 14.

  * (1)

If $u=\frac{p}{r}$ and $v=\frac{q}{r}$ for coprime integers $p,q,r$ such that
not all of $p,q,r$ are odd, then ${\cal F}^{2}(2,{D}_{\mathrm{S}}^{u,v})$ is a
null-set.

  * (2)

If $u$ or $v$ is irrational, then ${\cal F}^{2}(2,{D}_{\mathrm{S}}^{u,v})$ is
a null-set.

###### Proof.

(1) Suppose that $p,q,r$ are coprime and not all of $p,q,r$ are odd. Set
$a=-p$, $b=-q$, $c=r$. If $c$ is even then $a$ or $b$ is odd. On the other
hand, if $c$ is odd then $a$ or $b$ is even. Therefore, $(a,b,c)\in C$.
Therefore, by Lemma 13, $(a,b,c)\in{\cal
E}^{3}(2,\Delta(D_{\mathrm{S}^{\prime}}))$. Thus, by Lemma 12, ${\cal
F}(2,D_{\mathrm{S}^{\prime}})$ is projected to a null-set along $(a,b,c)$.
Along the vector $(a,b,c)$, ${\cal F}^{3}(2,D_{\mathrm{S}^{\prime}})$ is
projected to ${\cal F}^{2}(2,{D}_{\mathrm{S}}^{u,v})$ for
$u=-\frac{a}{c}=\frac{p}{r}$ and $v=-\frac{b}{c}=\frac{q}{r}$. Therefore,
${\cal F}^{2}(2,{D}_{\mathrm{S}}^{u,v})$ is a null-set.

(2) Suppose that $u$ or $v$ is irrational. Let $\varphi$ be the projection
along $(u,v,-1)$ that maps $D_{\mathrm{S}^{\prime}}$ to
${D}_{\mathrm{S}}^{u,v}$. We show that ${\cal
E}^{3}(2,\varphi({D}_{\mathrm{S}^{\prime}}))$ is not uniformly discrete. That
is,

$\displaystyle\forall\delta>0\ (\exists x,x^{\prime}\in{\cal
E}^{2}(2,\varphi({D}_{\mathrm{S}^{\prime}})))\ (\lvert
x-x^{\prime}\rvert\leq\delta\land x\neq x^{\prime})$
$\displaystyle\Leftrightarrow\ $ $\displaystyle\forall\delta>0\ \exists
x\in{\cal E}^{2}(2,\Delta(\varphi({D}_{\mathrm{S}^{\prime}})))\ (\lvert
x\rvert\leq\delta\land x\neq{\bf 0})$ $\displaystyle\Leftrightarrow\ $
$\displaystyle\forall\delta>0\ \exists x\in\varphi({\cal
E}^{3}(2,\Delta({D}_{\mathrm{S}^{\prime}}))\ (\lvert x\rvert\leq\delta\land
x\neq{\bf 0})$ $\displaystyle\Leftrightarrow\ $ $\displaystyle\forall\delta>0\
\exists y\in{\cal E}^{3}(2,\Delta({D}_{\mathrm{S}^{\prime}}))\
(\lvert\varphi(y)\rvert\leq\delta\land y\neq{\bf 0})$

The last equivalence holds because, for $y\in{\mathbb{Z}}^{3}$,
$\varphi(y)={\bf 0}$ iff $y={\bf 0}$. Let $\lfloor x\rfloor$ and $\\{x\\}$ be
the integer and fractional parts of $x\in{\mathbb{R}}$, respectively. We have
$\lvert\varphi((\lfloor 2iu\rfloor,\lfloor
2iv\rfloor,-2i))\rvert=\lvert\varphi((2iu-\\{2iu\\},2iv-\\{2iv\\},-2i))\rvert=\lvert(\\{2iu\\},\\{2iv\\})\rvert$.
If $u$ and $v$ are irrational, then $\lvert(\\{2iu\\},\\{2iv\\})\rvert<\delta$
for some $i$ because of the density of $\\{\\{2ix\\}\mid i\in{\mathbb{Z}}\\}$
for an irrational $x$. It is also the case if one of $u$ or $v$ is irrational
because $\\{i\mid\\{2ix\\}=0\\}$ is a progressive set if $x$ is rational.
Obviously, one can choose such $i$ so that $(\lfloor 2iu\rfloor,\lfloor
2iv\rfloor,-2i)\in{\cal E}^{3}(2,\Delta({D}_{\mathrm{S}^{\prime}}))$. ∎

###  5.2 T-fractal case

We consider the differenced radix expansion set of the following three-
dimensional digit set.

$\displaystyle
D_{\mathrm{T}^{\prime}}=\Bigl{\\{}\left(\\!\\!\begin{array}[]{r}1\\\ -1\\\
0\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}-1\\\ 1\\\
0\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}-1\\\ -1\\\
0\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}-1\\\ 0\\\
0\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}0\\\ -1\\\
0\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}0\\\ 0\\\
0\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}0\\\ 1\\\
1\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}1\\\ 0\\\
1\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}1\\\ 1\\\
1\end{array}\\!\\!\right)\Bigr{\\}}$

The sets ${D}_{\mathrm{T}^{\prime}}$ and $\Delta({D}_{\mathrm{T}^{\prime}})$
are depicted in Fig. 9. $D_{\mathrm{T}^{\prime}}$ is projected on the
$xy$-plane along $(a,b,c)$ to $D^{u,v}_{\mathrm{T}}$ for $u=1-\frac{a}{c}$ and
$v=1-\frac{b}{c}$. For $A\subseteq{\mathbb{R}}^{3}$ and $c\in{\mathbb{Z}}$, we
denote by $A_{c}$ the slice $\\{(a,b)\mid(a,b,c)\in A\\}$. One can see that

$\displaystyle\Delta({D}_{\mathrm{T}^{\prime}})_{1}$
$\displaystyle=\\{(i,j)\mid i,j<3\land i+j>0\\},$
$\displaystyle\Delta({D}_{\mathrm{T}^{\prime}})_{0}\ \ $
$\displaystyle=\\{(i,j)\mid\lvert i\rvert,\lvert j\rvert,\lvert
i+j\rvert<3\\},$ $\displaystyle\Delta({D}_{\mathrm{T}^{\prime}})_{-1}$
$\displaystyle=\\{(i,j)\mid i,j>-3\land i+j<0\\}.$

From the inductive definition of ${\cal
E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))$, we have

$\displaystyle{\cal E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))_{3n-1}$
$\displaystyle=3{\cal
E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))_{n}+\Delta({D}_{\mathrm{T}^{\prime}})_{-1}\,,$
$\displaystyle{\cal E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))_{3n}$
$\displaystyle=3{\cal
E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))_{n}+\Delta({D}_{\mathrm{T}^{\prime}})_{0}\,,$
$\displaystyle{\cal E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))_{3n+1}$
$\displaystyle=3{\cal
E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))_{n}+\Delta({D}_{\mathrm{T}^{\prime}})_{1}\,.$

Thus, ${\cal E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))_{z}$ can be described
through balanced trinary expansion of $z\in{\mathbb{Z}}$, which is the
sequence $(a_{i})_{0\leq i<t}$ for $a_{i}\in\\{-1,0,1\\}$ such that
$z=\Sigma_{i=0}^{t-1}a_{i}3^{i}$. ${\cal
E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))_{z}$ for $z=-1,0,1,2,3,6$ are shown in
Fig. 10. As these figures suggest, ${\cal
E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))$ has a bit complicated structure. We
only characterize ${\cal
E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))_{3^{m}(3^{n}-1)}$ for $m,n\geq 0$.We
define

$\displaystyle A_{m}$ $\displaystyle=\\{(i,j)\mid i,j\leq 0\land
i+j>=-3^{m+1}\\},$ $\displaystyle B$ $\displaystyle=3{\mathbb{Z}}^{2}.$

|
---|---
(a) $D_{\mathrm{T}^{\prime}}$ | (b) $\Delta(D_{\mathrm{T}^{\prime}})$
Figure 9:  (a) The digit set $D_{\mathrm{T}^{\prime}}$; (b) The differenced
digit set
$\Delta(D_{\mathrm{T}^{\prime}})={D}_{\mathrm{T}^{\prime}}+(-{D}_{\mathrm{T}^{\prime}})$.

Figure 10:  ${\cal E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))$ for $-14\leq x\leq
3$, $-11\leq y\leq 3$ and $z\in\\{-1,0,1,2,3,6\\}$. Points in $3{\cal
E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))$ are circled and translations of
$\Delta({D}_{\mathrm{T}^{\prime}})$ by
$(0,0,0),(3,-3,3),(0,3,3),(-3,0,3),(-6,-6,0)$ are colored red.

######  Lemma 15.

  1. (1)

${\cal E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))_{0}={\mathbb{Z}}^{2}$.

  2. (2)

${\cal
E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))_{3^{n}}={\mathbb{Z}}^{2}\setminus
3^{n}B$ for $n\geq 0$.

  3. (3)

${\cal
E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))_{3^{m}(3^{n}-1)}={\mathbb{Z}}^{2}\setminus
3^{m}B\setminus(3^{m+n}B+A_{m})$ for $m\geq 0$, $n>0$.

###### Proof.

(1) One can expand $(x,y)\in{\mathbb{Z}}^{2}$ as
$(x,y)=a_{t}3^{t}+\ldots+a_{0}$ for $a_{i}\in\\{-1,0,1\\}^{2}$. Since
$\\{-1,0,1\\}^{2}\subset\Delta({D}_{\mathrm{T}^{\prime}})_{0}$,
$(a_{i})_{i\geq 0}$ forms a
$(3,\Delta({D}_{\mathrm{T}^{\prime}})_{0})$-expansion of $(x,y)$.

(2) Induction on $n$. Case n = 0 : We have

$\displaystyle{\cal E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))_{1}$
$\displaystyle=3{\cal
E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))_{0}+\Delta({D}_{\mathrm{T}^{\prime}})_{1}=B+\Delta({D}_{\mathrm{T}^{\prime}})_{1}.$

It is equal to ${\mathbb{Z}}^{2}\setminus B$. See Fig. 10, $z=1$.

Case $n>0$: By induction hypothesis

$\displaystyle{\cal E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))_{3^{n}}$
$\displaystyle=3{\cal
E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))_{3^{n-1}}+\Delta({D}_{\mathrm{T}^{\prime}})_{0}$
$\displaystyle=3({\mathbb{Z}}^{2}\setminus
3^{n-1}B)+\Delta({D}_{\mathrm{T}^{\prime}})_{0}$ $\displaystyle=B\setminus
3^{n}B+\Delta({D}_{\mathrm{T}^{\prime}})_{0}.$

It is equal to ${\mathbb{Z}}^{2}\setminus 3^{n}B$. See Fig. 10, $z=3$.

(3) Induction on $m$. Case $m=0$: For $n>0$,

$\displaystyle{\cal E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))_{3^{n}-1}$
$\displaystyle=3{\cal
E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))_{3^{(n-1)}}+\Delta({D}_{\mathrm{T}^{\prime}})_{-1}$
$\displaystyle=3({\mathbb{Z}}^{2}\setminus
3^{n-1}B)+\Delta({D}_{\mathrm{T}^{\prime}})_{-1}$ $\displaystyle=B\setminus
3^{n}B+\Delta({D}_{\mathrm{T}^{\prime}})_{-1}.$

It is equal to ${\mathbb{Z}}^{2}\setminus B\setminus(3^{n}B+A_{0})$. See Fig.
10, $z=2$.

Case $m>0$: By induction hypothesis

$\displaystyle{\cal E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))_{3^{m}(3^{n}-1)}$
$\displaystyle=3{\cal
E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))_{3^{m-1}(3^{n}-1)}+\Delta({D}_{\mathrm{T}^{\prime}})_{0}$
$\displaystyle=3({\mathbb{Z}}^{2}\setminus
3^{m-1}B\setminus(3^{m-1+n}B+A_{m-1}))+\Delta({D}_{\mathrm{T}^{\prime}})_{0}$
$\displaystyle=B\setminus
3^{m}B\setminus(3^{m+n}B+3A_{m-1})+\Delta({D}_{\mathrm{T}^{\prime}})_{0}$

It is equal to ${\mathbb{Z}}^{2}\setminus 3^{m}B\setminus(3^{m+n}B+A_{m})$.
See Fig. 10, $z=6$.

∎

We prove the ’only if’ part of Theorem 6(b). In the proof of Lemma 14,
$(a,b,c)\in{\cal E}^{3}(k,\Delta(D_{\mathrm{S}^{\prime}}))$ for $a=-p$, $b=-q$
and $c=r$, and thus we could apply Lemma 12 for $m=1$. However, in the proof
of Lemma 16 below, $(a,b,c)\not\in{\cal
E}^{3}(k,\Delta(D_{\mathrm{T}^{\prime}}))$ in general and therefore the same
argument does not apply. Instead, we show that $(ma,mb,mc)\in{\cal
E}^{3}(k,\Delta(D_{\mathrm{T}^{\prime}}))$ for some $m>0$ to apply Lemma 12.

######  Lemma 16.

  * (1)

If $u=\frac{p}{r}$ and $v=\frac{q}{r}$ for coprime integers $p,q,r$ such that
$r>0$ and $\neg(p\equiv q\equiv r\mod 3)$, then ${\cal
F}^{2}(3,{D}_{\mathrm{T}}^{u,v})$ is a null-set.

  * (2)

If $u$ or $v$ is irrational, then ${\cal F}^{2}(3,{D}_{\mathrm{T}}^{u,v})$ is
a null-set.

###### Proof.

(1): Let $c=r$, $a=r-p$, $b=r-q$. Suppose that $p\equiv q\equiv r\mod 3$ does
not hold. Then, we have $\neg((3\mid a)\land(3\mid b))$. Let $m$ and
$c^{\prime}$ be $3^{m}c^{\prime}=c$ for $3\nmid c^{\prime}$. By Euler’s
theorem, $3^{\phi{(c^{\prime})}}\equiv 1\mod c^{\prime}$ for $\phi$ the
Euler’s function. Thus, for $n=\phi{(c^{\prime})}$ and some $s>0$, we have
$sc^{\prime}=3^{n}-1$ and thus $sc=3^{m}(3^{n}-1)$. Let $t_{0}=1$ and
$t_{k}=(3^{2^{k-1}n}+1)\ldots(3^{2n}+1)(3^{n}+1)$ for $k>0$. We have
$t_{1}sc=3^{m}(3^{n}+1)(3^{n}-1)=3^{m}(3^{2n}-1)$, and one can inductively
show $t_{k}sc=3^{m}(3^{2^{k}n}-1)$ for $k\geq 0$.

Suppose that $(t_{k}sa,t_{k}sb,t_{k}sc)\not\in{\cal
E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))$ for every $k\geq 0$. Then, by Lemma
15(3), $(t_{k}sa,t_{k}sb)\in(3^{m}B\cup(3^{m+2^{k}n}B+A_{m}))$. Since
$\neg((3\mid a)\land(3\mid b))$ and $3\nmid st_{k}$, we have
$(t_{k}sa,t_{k}sb)\not\in B$ and thus $(t_{k}sa,t_{k}sb)\not\in 3^{m}B$.
Therefore, $(t_{k}sa,t_{k}sb)\in 3^{m+2^{k}n}B+A_{m}$.

Let $j$ satisfy $2^{j}n\geq m+1$. We have $(t_{j}sa,t_{j}sb)\in
3^{m+2^{j}n}B+A_{m}$ and $(t_{j+1}sa,t_{j+1}sb)\in 3^{m+2^{j+1}n}B+A_{m}$,
which is impossible because
$\frac{t_{j+1}}{t_{j}}=3^{{2^{j}}n}+1>3^{m+1}=\mathrm{diam}(A_{m})$ for
$\mathrm{diam}(A_{m})$ the diameter of $A_{m}$.

Thus, $(t_{k}sa,t_{k}sb,t_{k}sc)\in{\cal
E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))$ for some $k$. Therefore, ${\cal
F}^{3}(3,D_{\mathrm{T}^{\prime}})$ is projected to a null-set along $(a,b,c)$
by Lemma 12.

(2): As in Lemma 14, we assume that $u$ or $v$ is irrational and prove

$\displaystyle\forall\l>0\ (\exists y\in{\cal
E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))\setminus\\{{\bf 0}\\})\
\lvert\varphi(y)\rvert\leq\frac{1}{3^{l}}\,.$

Let $\bm{u}=(u,v)\in{\mathbb{R}}^{2}$. For $a\in{\mathbb{R}}$, we write
$\lfloor a\rfloor$ and $\\{a\\}$ for the integer and fractional part $a$,
respectively, and for $\bm{a}=(a_{0},a_{1})\in{\mathbb{R}}^{2}$, we write
$\lfloor\bm{a}\rfloor$ and $\\{\bm{a}\\}$ for $(\lfloor a_{0}\rfloor,\lfloor
a_{1}\rfloor)$ and $(\\{a_{0}\\},\\{a_{1}\\})$, respectively. For $l>0$, we
divide $[0,1)^{2}$ into $3^{2l}$ disjoint regions
$r(s,t)=[\frac{s}{3^{l}},\frac{s+1}{3^{l}})\times[\frac{t}{3^{l}},\frac{t+1}{3^{l}})$
($0\leq s,t<3^{l}$). By the pigeonhole principle, there is a region $r(s,t)$
such that the set $P=\\{i\mid\\{3^{i}\bm{u}\\}\in r(s,t)\\}$ is infinite. For
the enumeration $0\leq p_{0}<p_{1}<\ldots$ of $P$, we define $h=p_{0}$,
$m=p_{1}$ and $n_{i}=p_{i+2}-m$ for $i=0,1,\ldots$.

Let $\bm{k}_{i}=\lfloor 3^{m+n_{i}}\bm{u}\rfloor-\lfloor 3^{m}\bm{u}\rfloor$,
$(a_{i},b_{i})=\bm{k}_{i}$, and $y_{i}=(a_{i},b_{i},3^{m}(3^{n_{i}}-1))$. For
each $i$, $y_{i}$ satisfies $\lvert\varphi(y_{i})\rvert\leq\frac{1}{3^{l}}$
because

$\displaystyle\lvert\varphi(y_{i})\rvert$
$\displaystyle=\lvert(a_{i},b_{i})-3^{m}(3^{n_{i}}-1)\bm{u}\rvert$
$\displaystyle=\lvert\bm{k}_{i}-3^{m+n_{i}}\bm{u}+3^{m}\bm{u}\rvert$
$\displaystyle=\lvert-\\{3^{m+n_{i}}\bm{u}\\}+\\{3^{m}\bm{u}\\}\rvert<\frac{1}{3^{l}}.$

Therefore, we complete the proof by showing that $y_{i}\in{\cal
E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))$ for some $i$. By Lemma 15, it means
$\bm{k}_{i}\in{\mathbb{Z}}^{2}\setminus 3^{m}B\setminus(3^{m+n_{i}}B+A_{m})$.
Therefore, we show that for some $i$, $\bm{k}_{i}\not\in 3^{m}B$ and
$\bm{k}_{i}\not\in 3^{m+n_{i}}B+A_{m}$.

Suppose that $\bm{k}_{i}\in 3^{m}B$ or $\bm{k}_{i}\in 3^{m+n_{i}}B+A_{m}$ for
every $i$. If

$\bm{k}_{i}=\lfloor 3^{m+n_{i}}\bm{u}\rfloor-\lfloor 3^{m}\bm{u}\rfloor\in
3^{m+n_{i}}B+A_{m}$

for infinite number of $i$. Then,

$3^{m+n_{i}}\bm{u}-3^{m}\bm{u}\in
3^{m+n_{i}}B+A_{m}+[-\frac{1}{3^{l}},\frac{1}{3^{l}}]^{2}.$

Therefore,

$\displaystyle\bm{u}$
$\displaystyle\in\frac{3^{n_{i}}}{3^{n_{i}}-1}B+\frac{1}{3^{n_{i}}-1}\frac{A_{m}+[-\frac{1}{3^{l}},\frac{1}{3^{l}}]^{2}}{3^{m}}$
$\displaystyle\subseteq\frac{3^{n_{i}}}{3^{n_{i}}-1}B+\frac{1}{3^{n_{i}}-1}[-1,4]^{2}.$

Since it holds for arbitrary large $n_{i}$ and the right-hand side converges
to $B$, we have $\bm{u}\in B$, but it contradicts the fact that at least one
component of $\bm{u}$ is irrational. Therefore, $\bm{k}_{i}\in
3^{m+n_{i}}B+A_{m}$ for a finite number of $i$, so let $I$ be their maximum.

We have $\bm{k}_{i}\in 3^{m}B$ for $i>I$. Let $(u_{j})_{j\geq 0}$ and
$(v_{j})_{j\geq 0}$ be the trinary expansions of the fractional parts of $u$
and $v$, respectively, and let $\sigma,\tau\in\\{0,1,2\\}^{l}$ be the trinary
expressions of $s$ and $t$, respectively. From the construction,
$(u_{j})_{j\geq 0}$ and $(v_{j})_{j\geq 0}$ contain $\sigma$ and $\tau$,
respectively, as subsequences starting at $h,m$ and $m+n_{i}$
($i=0,1,\ldots$).

Since $\bm{k}_{i}=\lfloor 3^{m+n_{i}}\bm{u}\rfloor-\lfloor
3^{m}\bm{u}\rfloor\in 3^{m}B$, the last $m$ digits of the trinary expansions
of $3^{m+n_{i}}u$ and $3^{m}u$ coincide. That is, $(u_{j})_{0\leq j<m}$ and
$(u_{j})_{n_{i}\leq j<m+n_{i}}$ coincide and thus $(u_{j})_{0\leq j<m+l}$ and
$(u_{j})_{n_{i}\leq j<m+n_{i}+l}$ coincide because $(u_{j})_{m\leq
j<m+l}=(u_{j})_{m+n_{i}\leq j<m+n_{i}+l}=\sigma$. Therefore, since
$(u_{j})_{0\leq j<m+l}$ contains $\sigma$ at $h$ and $m$, $(u_{j})_{n_{i}\leq
j<m+n_{i}+l}$ contains $\sigma$ at h + $n_{i}$ and $m+n_{i}$. It is also the
case for $(v_{j})_{j\geq 0}$ and $\tau$. Since $m+n_{i-1}$ is next to the last
index at which $\sigma$ and $\tau$ are contained in $(u_{j})_{0\leq
j<m+n_{i}+l}$ and $(v_{j})_{0\leq j<m+n_{i}+l}$ simultaneously, we have
$m+n_{i-1}=h+n_{i}$ and thus $n_{i}-n_{i-1}=m-h$ and $(u_{j})_{h\leq
j<m}=(u_{j})_{n_{i-1}\leq j<n_{i}}$ and $(v_{j})_{h\leq
j<m}=(v_{j})_{n_{i-1}\leq j<n_{i}}$. Since it holds for $i>I$, $u$ and $v$ are
rational numbers. It contradicts the fact that $u$ or $v$ is irrational. ∎

###  5.3 H-fractal case

---
(a) $D_{\mathrm{H}^{\prime}}$
(b) $\Delta(D_{\mathrm{H}^{\prime}})$
Figure 11:  (a) The digit set $D_{\mathrm{H}^{\prime}}$; (b) The differenced
digit set
$\Delta(D_{\mathrm{H}^{\prime}})={D}_{\mathrm{H}^{\prime}}+(-{D}_{\mathrm{H}^{\prime}})$.

Let

$\displaystyle
D_{\mathrm{H}^{\prime}}=\Bigl{\\{}\left(\\!\\!\begin{array}[]{r}0\\\ 0\\\
0\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}1\\\ 0\\\
0\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}1\\\ -1\\\
0\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}0\\\ -1\\\
0\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}-1\\\ 0\\\
0\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}-1\\\ 1\\\
0\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}0\\\ 1\\\
0\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}0\\\ 0\\\
1\end{array}\\!\\!\right)\\!,\left(\\!\\!\begin{array}[]{r}0\\\ 0\\\
-1\end{array}\\!\\!\right)\Bigr{\\}},$

which is projected on the $xy$-plane along $(a,b,c)$ to $D^{u,v}_{\mathrm{H}}$
for $u=-\frac{a}{c}$ and $v=-\frac{b}{c}$. The sets
${D}_{\mathrm{H}^{\prime}}$ and $\Delta({D}_{\mathrm{H}^{\prime}})$ are
depicted in Fig. 11.

We define

$\displaystyle A_{m}^{\prime}$ $\displaystyle=\\{(i,j)\mid(i,j\geq-3^{m}\land
i+j\leq 2(3^{m}))\lor(i,j\leq 3^{m}\land i+j\geq-2(3^{m}))\\},$ $\displaystyle
G_{m}^{\prime}$ $\displaystyle=\\{(i,j)\mid\lvert i\rvert,\lvert
j\rvert,\lvert i+j\rvert<3^{m}\\},$ $\displaystyle B^{\prime}$
$\displaystyle=\\{(i,j)\mid i\equiv j\not\equiv 0\mod 3\\}\
=3{\mathbb{Z}}^{2}+\\{(-1,-1),(1,1)\\}.$

We also define $J=\\{(i,j)\mid\lvert i\rvert,\lvert j\rvert,\lvert
i+j\rvert\leq 1\\}$. We have

$\displaystyle\Delta({D}_{\mathrm{H}^{\prime}})_{-2}$
$\displaystyle=\Delta({D}_{\mathrm{H}^{\prime}})_{2}=\\{(0,0)\\},$
$\displaystyle\Delta({D}_{\mathrm{H}^{\prime}})_{-1}$
$\displaystyle=\Delta({D}_{\mathrm{H}^{\prime}})_{1}=J,$
$\displaystyle\Delta({D}_{\mathrm{H}^{\prime}})_{0}\ \ $
$\displaystyle=G_{1}^{\prime}.$

From the inductive definition of ${\cal
E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))$, we have

$\displaystyle{\cal E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))_{3n-1}$
$\displaystyle=(3{\cal
E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))_{n}+\Delta({D}_{\mathrm{H}^{\prime}})_{-1})\cup(3{\cal
E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))_{n-1}+\Delta({D}_{\mathrm{H}^{\prime}})_{2})\,,$
$\displaystyle{\cal E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))_{3n}$
$\displaystyle=3{\cal
E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))_{n}+\Delta({D}_{\mathrm{H}^{\prime}})_{0}\,,$
$\displaystyle{\cal E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))_{3n+1}$
$\displaystyle=(3{\cal
E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))_{n}+\Delta({D}_{\mathrm{H}^{\prime}})_{1})\cup(3{\cal
E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))_{n+1}+\Delta({D}_{\mathrm{H}^{\prime}})_{-2})\,.$

######  Lemma 17.

  1. (1)

${\cal E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))_{0}={\mathbb{Z}}^{2}$.

  2. (2)

${\cal
E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))_{3^{n}}={\mathbb{Z}}^{2}\setminus
3^{n}B^{\prime}$ for $n\geq 0$.

  3. (3)

${\cal
E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))_{3^{m}(3^{n}-1)}=({\mathbb{Z}}^{2}\setminus
3^{m}B^{\prime}\setminus(3^{m+n}B^{\prime}+A_{m}^{\prime}))\cup(3^{m+n}B^{\prime}+G_{m}^{\prime})$
for $m\geq 0$, $n>0$.

###### Proof.

(1) The same as Lemma 15.

Figure 12:  ${\cal E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))$ for $-4\leq
x,y\leq 16,z\in\\{-1,0,1,2,3,6\\}$. Digits of $3{\cal
E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))$ are circled and translations of
$\Delta({D}_{\mathrm{H}^{\prime}})$ by $(0,0,0),(3,12,0),(9,6,3)$ are colored
red.

(2) Induction on $n$. Case n = 0 (see Fig. 12, z=1) : We have

$\displaystyle{\cal E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))_{1}$
$\displaystyle=(3{\cal
E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))_{0}+\Delta({D}_{\mathrm{H}^{\prime}})_{1})\cup(3{\cal
E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))_{1}+\Delta({D}_{\mathrm{H}^{\prime}})_{-2})$
$\displaystyle=3{\mathbb{Z}}^{2}+J={\mathbb{Z}}^{2}\setminus B^{\prime}.$

Case $n>0$ (see Fig. 12, z=3): By induction hypothesis

$\displaystyle{\cal E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))_{3^{n}}$
$\displaystyle=3{\cal
E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))_{3^{n-1}}+\Delta({D}_{\mathrm{H}^{\prime}})_{0}$
$\displaystyle=3({\mathbb{Z}}^{2}\setminus
3^{n-1}B^{\prime})+\Delta({D}_{\mathrm{H}^{\prime}})_{0}$
$\displaystyle=3{\mathbb{Z}}^{2}\setminus
3^{n}B^{\prime}+\Delta({D}_{\mathrm{H}^{\prime}})_{0}$
$\displaystyle={\mathbb{Z}}^{2}\setminus 3^{n}B^{\prime}\,.$

(3) We prove by induction on $m$. Case $m=0$ (see Fig. 12, z=2):

For $n=1$,

$\displaystyle{\cal E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))_{2}$
$\displaystyle=(3{\cal
E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))_{1}+\Delta({D}_{\mathrm{H}^{\prime}})_{-1})\cup(3{\cal
E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))_{0}+\Delta({D}_{\mathrm{H}^{\prime}})_{2})$
$\displaystyle=(3({\mathbb{Z}}^{2}\setminus B^{\prime})+J)\cup
3{\mathbb{Z}}^{2}$ $\displaystyle=(3{\mathbb{Z}}^{2}\setminus
3B^{\prime}+J)\cup(3{\mathbb{Z}}^{2}\setminus 3B^{\prime})\cup 3B^{\prime}$
$\displaystyle=(3{\mathbb{Z}}^{2}\setminus 3B^{\prime}+J)\cup 3B^{\prime}$
$\displaystyle={\mathbb{Z}}^{2}\setminus
B^{\prime}\setminus(3B^{\prime}+A_{0}^{\prime})\cup(3B^{\prime}+G_{0}^{\prime})$

For $n>1$,

$\displaystyle{\cal E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))_{3^{n}-1}$
$\displaystyle\hskip 28.45274pt=(3{\cal
E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))_{3^{(n-1)}}+\Delta({D}_{\mathrm{H}^{\prime}})_{-1})\cup(3{\cal
E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))_{3^{(n-1)}-1}+\Delta({D}_{\mathrm{H}^{\prime}})_{2})$
$\displaystyle\hskip 28.45274pt=(3({\mathbb{Z}}^{2}\setminus
3^{n-1}B^{\prime})+J)\cup 3({\mathbb{Z}}^{2}\setminus
B^{\prime}\setminus(3^{n-1}B^{\prime}+A_{0}^{\prime})\cup(3^{n-1}B^{\prime}+G_{0}^{\prime}))$
$\displaystyle\hskip 28.45274pt=(3{\mathbb{Z}}^{2}\setminus
3^{n}B^{\prime})+J)\cup 3{\mathbb{Z}}^{2}\setminus
3B^{\prime}\setminus(3^{n}B^{\prime}+3A_{0}^{\prime})\cup 3^{n}B^{\prime}$
$\displaystyle\hskip 28.45274pt=(3{\mathbb{Z}}^{2}\setminus
3^{n}B^{\prime})+J)\cup 3^{n}B^{\prime}$ $\displaystyle\hskip
28.45274pt=({\mathbb{Z}}^{2}\setminus
B^{\prime}\setminus(3^{n}B^{\prime}+A_{0}^{\prime}))\cup 3^{n}B^{\prime}.$

Case $m>0$ (see Fig. 12, z=6):

$\displaystyle{\cal E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))_{3^{m}(3^{n}-1)}$
$\displaystyle\hskip 28.45274pt=3{\cal
E}(3,\Delta({D}_{\mathrm{H}^{\prime}}))_{3^{m-1}(3^{n}-1)}+\Delta({D}_{\mathrm{H}^{\prime}})_{0}$
$\displaystyle\hskip 28.45274pt=3(({\mathbb{Z}}^{2}\setminus
3^{m-1}B^{\prime}\setminus(3^{m+n-1}B^{\prime}+A_{m-1}^{\prime}))\cup(3^{m+n-1}B^{\prime}+G_{m-1}^{\prime}))+G_{1}^{\prime}$
$\displaystyle\hskip 28.45274pt=(3{\mathbb{Z}}^{2}\setminus
3^{m}B^{\prime}\setminus(3^{m+n}B^{\prime}+3A_{m-1}^{\prime})\cup(3^{m+n}B^{\prime}+3G_{m-1}^{\prime}))+G_{1}^{\prime}$
$\displaystyle\hskip 28.45274pt=(3{\mathbb{Z}}^{2}\setminus
3^{m}B^{\prime}\setminus(3^{m+n}B^{\prime}+3A_{m-1}^{\prime})+G_{1}^{\prime})\cup(3^{m+n}B^{\prime}+3G_{m-1}^{\prime}+G_{1}^{\prime})$
$\displaystyle\hskip 28.45274pt=({\mathbb{Z}}^{2}\setminus
3^{m}B^{\prime}\setminus(3^{m+n}B^{\prime}+A_{m}^{\prime}))\cup(3^{m+n}B^{\prime}+G_{m}^{\prime}).$

∎

Now, we prove the ’only if’ part of Theorem 6(c):

######  Lemma 18.

  * (1)

If $u=\frac{p}{r}$ and $v=\frac{q}{r}$ for coprime integers $p,q,r$ such that
$r>0$ and $\neg(p\equiv q\equiv r\mod 3)\land\neg(p\equiv q\equiv-r\mod 3)$,
then ${\cal F}^{2}(3,{D}_{\mathrm{H}}^{u,v})$ is a null-set.

  * (2)

If $u$ or $v$ is irrational, then ${\cal F}^{2}(3,{D}_{\mathrm{H}}^{u,v})$ is
a null-set.

###### Proof.

(1): Let $c=r$, $a=-p$, $b=-q$ and $3^{m}c^{\prime}=c$ for $3\nmid
c^{\prime}$. Suppose that neither $p\equiv q\equiv r\mod 3$ nor $p\equiv
q\equiv-r\mod 3$ hold. Then, $m>0\land\neg((3\mid a)\land(3\mid b))$ or
$m=0\land\neg(a\equiv b\not\equiv 0\mod 3)$. We define $s,n,t_{k}(k\geq 0)$ as
in the proof of Lemma 16. We have $t_{k}sc=3^{m}(3^{2^{k}n}-1)$ for $k\geq 0$.
We show that $(t_{k}sa,t_{k}sb,t_{k}sc)\in{\cal
E}^{3}(3,\Delta({D}_{\mathrm{H}^{\prime}}))$ for some $k\geq 0$.

Suppose that $(t_{k}sa,t_{k}sb,t_{k}sc)\not\in{\cal
E}^{3}(3,\Delta({D}_{\mathrm{H}^{\prime}}))$ for every $k\geq 0$. By Lemma
17(3),
$(t_{k}sa,t_{k}sb)\in(3^{m}B^{\prime}\cup(3^{m+2^{k}n}B^{\prime}+A_{m}^{\prime}))$
and $(t_{k}sa,t_{k}sb)\not\in(3^{m+2^{k}n}B^{\prime}+G_{m}^{\prime})$. If
$m=0$, then $\neg(a\equiv b\not\equiv 0\mod 3)$. That is, $(a,b)\not\in
B^{\prime}$. In addition, $t_{k}s\not\equiv 0\mod 3$. Therefore,
$(t_{k}sa,t_{k}sb)\not\in B^{\prime}$. If $m>0$, then $\neg((3\mid
a)\land(3\mid b))$. Therefore, $\neg((3\mid t_{k}sa)\land(3\mid t_{k}sb))$ and
thus $(t_{k}sa,t_{k}sb)\not\in 3^{m}B^{\prime}$. In both cases,
$(t_{k}sa,t_{k}sb)\not\in 3^{m}B^{\prime}$. Therefore, $(t_{k}sa,t_{k}sb)\in
3^{m+2^{k}n}B^{\prime}+A_{m}^{\prime}$.

The rest of the proof is the same as that of Lemma 16 (1) with $A_{m}$ and $B$
substituted to $A_{m}^{\prime}$ and $B^{\prime}$, respectively.

(2): As in Lemma 16, we assume that $u$ or $v$ is irrational and prove

$\displaystyle\forall\l>0\ (\exists y\in{\cal
E}(3,\Delta({D}_{\mathrm{T}^{\prime}}))\setminus\\{{\bf 0}\\})\
\lvert\varphi(y)\rvert\leq\frac{2}{3^{l}}\,.$

The proof is essentially the same. Note that $\bm{k}_{i}\in 3^{m}B^{\prime}$
also means that the last $m$ digits of the trinary expansions of
$3^{m+n_{i}}u$ and $3^{m}u$ coincide.

∎

##  6 Concluding remarks

We characterized directions from which Sierpinski tetrahedron, T-fractal, and
H-fractal are projected to sets with positive measures.

A natural question is whether our result can be generalized to fractal
imaginary cubes of degree $\geq 4$. According to [14], there are 36 non-
congruent fractal imaginary cubes of degree 4 (including Sierpinski
tetrahedron that already appeared at degree 2), and 3482 fractal imaginary
cubes of degree 5. Since digit sets of fractal imaginary cubes of degree $k$
exist corresponding to Latin squares of size $k$ and it is shown in [17] that
the number of Latin squares of size $k$ is lower bounded by
$(k!)^{2k}/k^{k^{2}}$ the number of fractal imaginary cubes increases rapidly.
Among the 36 fractal imaginary cubes of degree 4, some have positively-
measured symmetric projected images along the line $x=y=z$ ([14]), but some do
not. We only have the following simple observation about fractal imaginary
cubes in general.

######  Proposition 19.

A degree-$k$ fractal imaginary cube ${\cal F}(k,D)$ of the unit cube
$[0,1]^{3}$ is projected to a set with positive measure along $(nk,mk,1)$ for
integers $m,n,k$, and their permutations.

###### Proof.

By Theorem 3, $D$ satisfies

$\\{(x,y)\mid(x,y,z)\in D\\}=\\{0,\ldots,k-1\\}^{2}\,.$

Therefore, the projected image of $D$ to the $xy$-plane, which is
$\\{(x-nkz,y-mkz)\mid(x,y,z)\in D\\}$, is a complete residue system of
${\mathbb{Z}}^{2}/k{\mathbb{Z}}\times k{\mathbb{Z}}$. The statement holds by
Corollary 10. ∎

Among fractal imaginary cubes, H-fractal has the distinct property that it is
a fractal double imaginary cube, and this additional symmetricity ’doubles’
the number of projected images with positive measure compared to the case of
T-fractal. It is shown in [15] that all convex double imaginary cubes are
variants of H, and one can easily observe from this fact that the convex hull
of a fractal double imaginary cube is always the polyhedron H. It would be
interesting to know whether H-fractal is the only fractal double imaginary
cube.

As related works, Kenyon studied in [7] line-projections of the one-
dimensional Sierpinski gasket and characterized directions from which it is
projected to sets with positive measures. It is interesting to know whether
one can obtain his result also through the method presented in this paper.
Hochman showed that Hausdorff dimensions of irrational projections of the one-
dimensional Sierpinski gasket are zero in [4], and Hausdorff dimension of
projected images of fractals are surveyed in [3] and [13]. Hausdorff
dimensions of the projected images of fractals studied in this paper are left
open.

Acknowledgement The author would like to thank Takayuki Tokuhisa for
illuminating discussions.

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##  Appendix A Pictures of a H-fractal model

##  Appendix B Pictures of a T-fractal model