# An Investigation of Multiplicative Invariance in the Complex Plane

Neil MacVicar

###### Abstract.

Multiplicative invariance is a well-studied notion in the unit interval. The
picture in the complex plane is less developed. This document introduces an
analogous notion of multiplicative invariance in the complex plane and
establishes similar results of Furstenberg’s from [3] in this setting. Namely,
that the Hausdorff and box-counting dimensions of a multiplicatively invariant
subset are equal and, furthermore, are equal to the normalized topological
entropy of an underlying subshift. We also extend a formula from [10] for the
box-counting dimension of base-$b$ restricted digit sets where $b$ is a
suitably chosen Gaussian integer.

## Introduction

Throughout his career, Abel prize winner Hillel Furstenberg made contributions
to many areas of mathematics using dynamical methods. Among those
contributions is a pair of papers at the intersection of dynamics and fractal
geometry ([3], [4]). Therein, Furstenberg proved results and stated
conjectures about the fractal properties of multiplicatively invariant subsets
of the unit interval. Multiplicatively invariant subsets are those that are
invariant under the map $x\mapsto rx\mod 1$ where $r$ is some positive
integer. For a specific value $r$, this is called $\times r$-invariance. The
following theorem highlights particular results of Furstenberg which are
recalled in Section 1 of this paper.

######  Theorem 0.1 (H. Furstenberg, [3], proposition III.1).

Let $r\geq 2$ be an integer. Let $\mathcal{E}$ denote topological entropy, let
$\dim_{H}$ denote Hausdorff dimension, and let $\dim_{B}$ denote box-counting
dimension. If $A\subset\\{0,1,\ldots,r-1\\}^{\mathbb{N}}$ is a subshift, then

  * (i)

$\tilde{A}=\\{\sum_{k=1}^{\infty}a_{k}r^{-k}:(a_{k})_{k\geq 1}\in A\\}$ is
$\times r$-invariant,

  * (ii)

$\dim_{B}\tilde{A}=\frac{\mathcal{E}(A)}{\log{r}}.$

  * (iii)

If $X$ is a $\times r$-invariant set, then $\dim_{H}X=\dim_{B}X.$

Considerable development of the theory of multiplicatively invariant subsets
of the unit interval has been pursued since: Furstenberg’s sumset conjecture,
which offers sufficient conditions under which the Hausdorff and box-counting
dimensions of sumsets of multiplicative invariant subsets split into the sum
of the dimensions of those subsets, was proven by Hochman and Shmerkin in [7]
(2012). Additionally, Furstenberg’s intersection conjecture (now known as the
Shmerkin-Wu theorem which implies the sumset conjecture) was proven
independently by Shmerkin in [11] (2019) and Wu in [13] (2019) using different
methods and again by Austin in [1] (2022).

In [6] (2021), Richter, Moreira, and Glasscock established similar results to
those of Furstenberg in [3] and a sumset result for a version of $\times
r$-invariance observed for subsets of the nonnegative integers.

The picture in the complex plane is less developed. What has been considered
by Pedersen and Shaw in [10] (2021) is a complex analogue of a class of
multiplicative invariant subsets called base-$r$ restricted digit Cantor sets.
A base-$r$ restricted digit Cantor set contains those numbers in the unit
interval that, when written in base-$r$, restrict the coefficients used in
their expansions to some subset of $\\{0,1,\ldots,r-1\\}$. For example, the
middle-thirds Cantor set are all numbers in the unit interval that, when
written in base $3$, only use the coefficients $0$ and $2$.

In this document we consider a complex analogue to the class of
multiplicatively invariant subsets. The problem of defining a more general
class of sets that might be called “$\times b$-invariant” where $b$ is a
Gaussian integer presents challenges that differ from the real case. The map
used to define multiplicative invariance in the unit interval subtracts the
integer part to ensure that the image is in the domain. It is not immediately
clear what the correct choice is for the integer part of a complex number.
This is complicated by a fact from [5] by Gilbert that we can have up to three
expansions in base-$b$ for the same complex number.

This document introduces a notion of $\times b$-invariance when $b$ is a
suitably chosen Gaussian integer (Definition 2.10). We show that $\times
b$-invariant sets share properties with their real counterparts. Namely, we
establish the following results which are similar to Theorem 0.1 (See Theorem
2.22 and Theorem 2.26).

######  Theorem 0.2.

Let $b=-n+i$ where $n\geq 3$ and assume $D\subset\\{0,1,\ldots,|b|^{2}-1\\}$
is separated. Let $g:D^{\mathbb{N}}\rightarrow C_{D}$ be the map
$(d_{j})_{j\geq 1}\mapsto\sum_{j\geq 1}d_{j}b^{-j}$. If $A\subset
D^{\mathbb{N}}$ is a subshift, then

  * (i)

$g(A)=\\{\sum_{k=1}^{\infty}a_{k}b^{-k}:(a_{k})_{k\geq 1}\in A\\}$ is $\times
b$-invariant,

  * (ii)

$\dim_{B}g(A)=\frac{\mathcal{E}(A)}{\log{|b|}}.$

  * (iii)

If $Y\subset C_{D}$ is a $\times b$-invariant set, then $\dim_{H}Y=\dim_{B}Y.$

In addition to this, we extend the application of a formula for box-counting
dimensions of base-$b$ restricted digit Cantor sets presented in [10] (see
Theorem 2.15).

######  Theorem 0.3.

Let $b=-n+i$ where $n\geq 2$ and suppose $D$ is a subset of $\Lambda_{b}$.
Then

(1) $\dim_{B}C_{D}=\frac{\log{|D|}}{\log{|b|}}.$

The statement in [10] assumes and is proved for the case when every pair of
distinct elements of $D$ is at least distance $n+1$ apart.

## Organization

This document is separated into two sections and two short appendices.

  1. (1)

Section 1 reviews the basics of multiplicative invariance in the unit interval
and includes concepts from fractal geometry and symbolic dynamics that are
used in Section 2.

  2. (2)

Section 2 includes the facts about base-$b$ expansions used to formulate
complex multiplicative invariance (Definition 2.10) and the proof of Theorem
0.2 (Theorem 2.22 and Theorem 2.26).

  3. (A)

Appendix A illustrates the derivation of the rules governing base-$(-n+i)$
expansions when $n\geq 3$.

  4. (B)

Appendix B includes the rules governing the special case of base-$(-2+i)$
expansions.

##  1\. Multiplicative Invariance in $\mathbb{R}$

In this section we recall the notion of multiplicative invariance for subsets
of the unit interval. We review the fractal properties of these subsets that
we wish to replicate in the complex plane.

######  Definition 1.1.

Let $r$ be a positive integer. Define the map

$T_{r}:\mathbb{R}\rightarrow[0,1)$ $x\mapsto rx\mod{1}.$

A nonempty closed subset $X\subset[0,1]$ is called $\times r-$invariant if
$T_{r}(X)\subset X$. A subset $X$ is multiplicatively invariant if it is
$\times r$-invariant for some $r\geq 2$.

######  Example 1.2.

Let $r$ be a positive integer. Suppose $D$ is a subset of
$\Lambda_{r}:=\\{0,1,\ldots,r-1\\}$. We call the set

(2) $C_{r,D}:=\bigg{\\{}x=\sum_{k=1}^{\infty}d_{k}r^{-k}\in\mathbb{R}:d_{k}\in
D\bigg{\\}}$

the base-$r$ restricted digit Cantor set with digit set $D$. These sets are
$\times r$-invariant.

The fractal properties of multiplicatively invariant sets are expressed
through their Hausdorff and box-counting dimensions. We review these
dimensions here.

######  Definition 1.3.

Let $\delta>0$ and $F\subset\mathbb{R}^{n}$. A countable collection of sets
$\\{U_{k}\\}$ is called a $\delta$-cover of $F$ if

  1. (i)

$F\subset\bigcup_{k}U_{k}$,

  2. (ii)

$\operatorname{diam}{U_{k}}\leq\delta$ for each $k$.

######  Definition 1.4.

Let $F\subset\mathbb{R}^{n}$ and let $s>0$. For every $\delta>0$, define the
quantity

(3)
$\mathcal{H}_{\delta}^{s}(F):=\inf{\bigg{\\{}\sum_{k}(\operatorname{diam}{U_{k}})^{s}:\\{U_{k}\\}\>\text{is
a $\delta$-cover of}\>F\bigg{\\}}}.$

The $s$-dimensional Hausdorff measure of $F$ is the limiting value
$\mathcal{H}^{s}(F):=\lim_{\delta\rightarrow
0^{+}}\mathcal{H}_{\delta}^{s}(F)$.

We call the quantity

(4) $\dim_{H}{F}:=\inf{\\{s\geq 0:\mathcal{H}^{s}(F)=0\\}}$

the Hausdorff dimension of $F$.

The Hausdorff dimension can be equivalently defined using less general covers.
For example, it is common to add the additional condition that the
$\delta$-covers only contain balls.

######  Proposition 1.5 (K. Falconer, [2], section 2.4).

Let $F\subset\mathbb{R}^{n}$ and define

(5)
$\mathcal{B}_{\delta}^{s}(F):=\inf{\bigg{\\{}\sum_{k}(\operatorname{diam}{B_{k}})^{s}:\\{B_{k}\\}\>\text{is
a $\delta$-cover of}\>F\>\text{by balls}\bigg{\\}}}.$

Then $\dim_{H}{F}:=\inf{\\{s\geq 0:\mathcal{B}^{s}(F)=0\\}}$ where
$\mathcal{B}^{s}(F)=\lim_{\delta\rightarrow
0^{+}}\mathcal{H}_{\delta}^{s}(F)$.

The Hausdorff dimension exhibits desirable properties, but it is difficult to
compute directly. The box-counting dimension is a popular alternative because
of the comparative ease of computing it.

######  Definition 1.6.

Let $F\subset\mathbb{R}^{n}$ be bounded. Let $\delta>0$. Let $N_{\delta}(F)$
denote the minimum number of subsets of $\mathbb{R}^{n}$ of diameter at most
$\delta$ required to cover $F$. If it exists, we call the limit

(6) $\dim_{B}F:=\lim_{\delta\rightarrow
0^{+}}\frac{\log{N_{\delta}(F)}}{-\log{\delta}}$

the box-counting dimension of $F$.

In the event the limit does not exist, we refer to the upper and lower limits
of the above function of $\delta$ as the upper and lower box-counting
dimensions respectively. This fractal dimension is useful because the
$N_{\delta}$ function has several equivalent formulations (see [2] for a
list). We will use an equivalent formulation found in [10] in the next
section.

Multiplicatively invariant subsets of the unit interval are also connected to
subshifts. We review the relevant definitions.

######  Definition 1.7.

Let $V$ be a finite set equipped with the discrete topology. Let
$\Omega=V^{\mathbb{N}}$ be the sequence space equipped with the product
topology and define the left shift map

$\sigma:\Omega\rightarrow\Omega$ $(v_{k})_{k\geq 1}\mapsto(v_{k+1})_{k\geq
1}.$

We call $A\subset X$ a subshift if it is closed and satisfies
$\sigma(A)\subset A$.

######  Definition 1.8.

Let $A$ be a subshift. The topological entropy of $A$ is the limit

(7)
$\mathcal{E}(A):=\lim_{n\rightarrow\infty}\frac{\log|\mathcal{L}_{n}(A)|}{n}$

where
$\mathcal{L}_{n}(A):=\\{(a_{1},a_{2},\ldots,a_{n}):a_{1}=\omega_{1},\ldots,a_{n}=\omega_{n}\;\textit{for
some}\;(\omega_{k})_{k\geq 1}\in A\\}$.

We remark that a more general definition of topological entropy can be found
in chapter 7 section 1 of [12] for continuous maps acting on compact spaces.
This more general notion is shown in theorem 7.13 of [12] to reduce to the
formula above in the case of subshifts and, in particular, the limit exists.

We now state a result of Furstenberg’s ([3], proposition III.1) about
multiplicatively invariant subsets of $[0,1]$ in two parts. We will formulate
these results for subsets of $\mathbb{C}$ in the next section.

######  Theorem 1.9 (H. Furstenberg, [3], proposition III.1).

Let $r\geq 2$ be an integer. If $A\subset\Lambda_{r}^{\mathbb{N}}$ is a
subshift, then

  * (i)

$\tilde{A}=\\{\sum_{k=1}^{\infty}a_{k}r^{-k}:(a_{k})_{k\geq 1}\in A\\}$ is
$\times r$-invariant,

  * (ii)

$\dim_{B}\tilde{A}=\frac{\mathcal{E}(A)}{\log{r}}.$

######  Theorem 1.10 (H. Furstenberg, [3], proposition III.1).

Let $X$ be a $\times r$-invariant set. Then $\dim_{H}X=\dim_{B}X.$

######  Remark 1.11.

In [3], proposition III.1 includes the assertion that the Hausdorff and box-
counting dimensions of the set $\tilde{A}$ in Theorem 1.9 are equal. In the
context of that statement, the set $\tilde{A}$ is the image of the subshift
$A$ under the map $(x_{k})_{k\geq 1}\mapsto\sum_{k=1}^{\infty}x_{k}r^{-k}$.
Observe that the preimage of a $\times r$-invariant set is a subshift of
$\Lambda_{r}^{\mathbb{N}}$ and hence we can claim the equality for Hausdorff
and box-counting dimensions for all $\times r$-invariant sets.

######  Example 1.12.

The middle-third Cantor set is the image of the set of sequences
$\\{(a_{k})_{k\geq 1}:a_{k}\in{0,2}\\}$ under the map $f$ in Theorem 1.9. The
topological entropy of this subshift according to Definition 1.8 is $\log{2}$.
It follows from the previous two theorems that
$\dim_{H}C_{3,\\{0,2\\}}=\dim_{B}C_{3,\\{0,2\\}}=\log{2}/\log{3}$.

It is natural to ask if claims of this kind hold for subsets of the complex
plane. The next section addresses this question.

##  2\. Multiplicative Invariance in $\mathbb{C}$

An important class of multiplicatively invariant subsets of the unit interval
are the restricted digit Cantor sets. This section introduces an analogous
class of subsets of the complex plane and then develops a notion of
multiplicative invariance. We begin with a result in [9] which provides
conditions on a Gaussian integer $b$ to ensure that any complex number to be
written in base $b$ with coefficients in $\\{0,1,\ldots,|b|^{2}-1\\}$.

######  Theorem 2.1 (I. Katai, J. Szabo, [9], theorem 2).

Suppose $n$ is a positive integer and set $b=-n+i$. Let $z$ be an element of
$\mathbb{C}$. There exist coefficients
$d_{k}\in\Lambda_{b}:=\\{0,1,\ldots,|b|^{2}-1\\}$ and some integer $l$ such
that

(8)
$z=d_{\ell}b^{\ell}+d_{\ell-1}b^{\ell-1}+\cdots+d_{0}+\sum_{k=1}^{\infty}d_{-k}b^{-k}.$

We will continue to use $\Lambda_{b}$ to denote the full coefficient set
throughout the remainder of this document. We remark that this is a slight
abuse of notation from Section 1. The set $\Lambda_{b}$, where $b=-n+i$, is
interpreted differently than $\Lambda_{r}$ where $r$ is a positive integer.

The expansions in the previous theorem are called radix expansions. It is
convenient to use the notation

(9) $(d_{\ell},d_{\ell-1},\ldots,d_{0};d_{-1},\ldots)$

for the expansion. The base $b$ is implicit in this notation because we only
consider a single base $b=-n+i$ in any of the discussions that follow. We use
the notation $d_{\ell}d_{\ell-1}\cdots d_{0}.d_{-1}\cdots$ to denote the
complex number represented by (9). The point that we would call the decimal
point, if this was an expansion in base ten, is called the radix point. We
will refer to the digits to the left of the radix point
$(d_{\ell},d_{\ell-1},\ldots d_{0};)$ as the integer part of the expansion.
The complex number represented by the integer part of a radix expansion is the
Gaussian integer $d_{\ell}b^{\ell}+d_{\ell-1}b^{\ell-1}+\cdots+d_{0}$.

######  Definition 2.2.

Let $b=-n+i$ where $n$ is a positive integer. Suppose $D$ is a subset of
$\Lambda_{b}$. We call the set

(10) $C_{D}:=\bigg{\\{}z=\sum_{k=1}^{\infty}d_{k}b^{-k}\in\mathbb{C}:d_{k}\in
D\bigg{\\}}$

the base-$b$ restricted digit Cantor set with digit set $D$.

We again omit any indication of the base $b=-n+i$ for the same reason the base
$b$ is implicit in the notation for radix expansions.

######  Remark 2.3.

We do explicitly prove that $C_{D}$ is a Cantor space given a condition on $D$
in Lemma 2.19 and Corollary 2.20. It is less work, however, to see that
$C_{D}$ is compact for any subset $D$ of $\Lambda_{b}$. Observe that
$D^{\mathbb{N}}$ is compact with the product topology. The evaluation map
given by $(d_{k})_{k\geq 1}\mapsto\sum_{k\geq 1}d_{k}b^{-k}$ is a continuous
map onto $C_{D}$. The compactness of $C_{D}$ tells us that it is the unique
attractor of the iterated function system $\\{z\mapsto\frac{z+d}{b}:d\in D\\}$
(invariance under the Hutchinson operator can be verified directly, see [8]).

Radix expansions of complex numbers, like expansions of real numbers in an
integer base, are not unique. In fact, it is shown in [5] that there can be as
many as three different radix expansions in the same base for the same complex
number. A result of Gilbert in [5] places conditions on a pair of equivalent
radix expansions. We require the following notation to state it.

Let $p=(p_{\ell},p_{\ell-1},\ldots,p_{0};p_{-1},\ldots)$ be a radix expansion
and let $k$ be an integer. We denote the Gaussian integer represented by the
integer part of the radix expansion
$(p_{\ell},p_{\ell-1},\ldots,p_{k};p_{k-1},\ldots)$ by $p(k)$.

######  Lemma 2.4 (W. J. Gilbert, [5], proposition 1).

Let $n$ be a postive integer. Two radix expansions, $q$ and $r$, represent the
same complex number in base $b=-n+i$ if and only if, for all integers $k$,
either

  * (i)

$q(k)-r(k)\in\\{0,\pm 1,\pm(n+i),\pm(n-1+i)\\}$ when $n\neq 2$, or

  * (ii)

$q(k)-r(k)\in\\{0,\pm 1,\pm(2+i),\pm(1+i),\pm i,\pm(2+2i)\\}$ when $n=2$.

This restriction can be used to deduce what expansions are possible for
complex numbers that have multiple radix expansions. It is also through this
analysis that it can be shown that a complex number has at most three
representations in base $b=-n+i$. We restrict ourselves to the case that
$n\geq 2$. The non-trivial subsets of the digit set $\Lambda_{-1+i}$ cause
$C_{D}$ to be a singleton.

In [5], Gilbert derives a state graph that governs triples of radix expansions
that represent the same complex number. We present the exposition used to
derive and parse the graph.

Suppose $p,q$ and $r$ are radix expansions of the same complex number. We do
not assume that they are distinct. We define the $k$th state of $p,q$ and $r$
to be the triple

(11) $S(k):=(p(k)-q(k),q(k)-r(k),r(k)-p(k)).$

Notably, since the sum of these components is zero, one of the components is
redundant. Nonetheless, it is useful to express all the differences explicitly
in order to determine the digits at the $k$th place of the expansions $p,q,$
and $r$. We describe this now.

If $p=(p_{\ell},p_{\ell-1},\ldots p_{0};p_{-1},\ldots)$, then $p(k+1)$ is the
Gaussian integer with radix expansion $(p_{\ell},p_{\ell-1},\ldots,p_{k+1};)$.
Therefore we have $p(k)=bp(k+1)+p_{k}$. It follows that
$p(k)-q(k)=p_{k}-q_{k}+b(p(k+1)-q(k+1))$. We can capture this as a
relationship between states with the equation

(12) $S(k)=(p_{k}-q_{k},q_{k}-r_{k},r_{k}-p_{k})+bS(k+1).$

Therefore the knowledge of the value of $S(k+1)$ can be used with Lemma 2.4 to
determine the possible values for the digits $p_{k},q_{k}$ and $r_{k}$ and the
state $S(k)$.

If we treat allowable states as nodes, we can contruct the state graph
presented in [5]. The directed edges indicate what states $S(k)$ can be
achieved from a given state $S(k+1)$ (the node you are currently at). The
graph in Figure 1 corresponds to the cases $n\geq 3$ where $b=-n+i$. The case
$n=2$ is more complicated and is presented in Appendix B. Both graphs feature
a system of diagrams that communicate the value of a state. We describe the
system for the case $n\geq 3$ here. The additional states present in the case
$n=2$ can be found in Appendix B.

We begin with a system of diagrams that communicate the value of $p(k)-q(k)$.
The system is as follows:

  1. (i)

$p(k)-q(k)=0$ corresponds to pq.

  2. (ii)

$p(k)-q(k)=1$ corresponds to qp.

  3. (iii)

$p(k)-q(k)=n-1+i$ corresponds to pq.

  4. (iv)

$p(k)-q(k)=n+i$ corresponds to qp.

pqrpqrpqrrpqrpqrpqpqrprqqprrqprpqpqrqrp$\scriptsize\begin{matrix}0\\\ 0\\\
0\\\ \end{matrix}$+$\scriptsize\begin{matrix}0\\\ 0\\\ 1\\\
\end{matrix}$+$\scriptsize\begin{matrix}1\\\ 1\\\ 0\\\
\end{matrix}$+$\scriptsize\begin{matrix}1\\\ 0\\\ 2n\\\
\end{matrix}$+$\scriptsize\begin{matrix}2n-1\\\ 2n\\\ 0\\\
\end{matrix}$+$\scriptsize\begin{matrix}0\\\ 1\\\ n^{2}-2n+2\\\
\end{matrix}$+$\scriptsize\begin{matrix}n^{2}-2n+2\\\ n^{2}-2n+1\\\ 0\\\
\end{matrix}$+$\scriptsize\begin{matrix}2n-1\\\ 0\\\ n^{2}\\\
\end{matrix}$$\scriptsize\begin{matrix}n^{2}\\\ 2n-1\\\ 0\\\
\end{matrix}$$\scriptsize\begin{matrix}0\\\ n^{2}\\\ 2n-1\\\
\end{matrix}$$\scriptsize\begin{matrix}n^{2}-2n+1\\\ n^{2}\\\ 0\\\
\end{matrix}$$\scriptsize\begin{matrix}0\\\ n^{2}-2n+1\\\ n^{2}\\\
\end{matrix}$$\scriptsize\begin{matrix}n^{2}\\\ 0\\\ n^{2}-2n+1\\\
\end{matrix}$$\scriptsize\begin{matrix}0\\\ 0\\\ n^{2}-2n+1\\\
\end{matrix}$+$\scriptsize\begin{matrix}0\\\ 0\\\ 2n-1\\\
\end{matrix}$+$\scriptsize\begin{matrix}0\\\ 0\\\ 2n\\\
\end{matrix}$+$\scriptsize\begin{matrix}2n-1\\\ 2n-1\\\ 0\\\
\end{matrix}$+$\scriptsize\begin{matrix}0\\\ 0\\\ n^{2}-2n+2\\\
\end{matrix}$+$\scriptsize\begin{matrix}n^{2}-2n+2\\\ n^{2}-2n+2\\\ 0\\\
\end{matrix}$+$\scriptsize\begin{matrix}2n\\\ 2n\\\ 0\\\
\end{matrix}$+$\scriptsize\begin{matrix}n^{2}-2n+1\\\ n^{2}-2n+1\\\ 0\\\
\end{matrix}$+$\scriptsize\begin{matrix}0\\\ 0\\\ n^{2}\\\
\end{matrix}$$\scriptsize\begin{matrix}n^{2}\\\ n^{2}\\\ 0\\\ \end{matrix}$
Figure 1. The graph governing equivalent radix expansions in base $-n+i$ for
$n\geq 3$.

Swapping the positions of $p$ and $q$ in any of these arrangements flips the
sign on the value of $p(k)-q(k)$. We can use this system to represent the
mutual differences between $p(k),q(k)$ and $r(k)$ simultaneously. For example,
the state $(1,-n-i,n-1+i)$ is communicated by

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Each edge of the state graph is labelled with a triple of integers. These
indicate a combination of digits, read from top to bottom, that $p_{k}$,
$q_{k}$, and $r_{k}$ can be in order for (12) to hold. The indication of a
“$+$” symbol means that we may add the integer $t$ to each of the values,
where $t$ can be $0,1,\ldots$ up to the largest integer for which all three of
the listed numbers, when shifted by $t$, are less than or equal to
$n^{2}=|b|^{2}-1$. Therefore the integers listed along the edges in the state
graph communicate the distances between the digits at that index.

######  Theorem 2.5 (W. J. Gilbert, [5], theorem 5).

Let $p,q$ and $r$ be three radix expansions in base $-n+i$ where $n\geq 3$.
These expansions represent the same complex number if and only if they can be
obtained from an infinite path through the state graph in Figure 1 starting at
state $(0,0,0)$, if necessary relabelling $p,q$ and $r$.

We include the derivation of figure 1 in the appendix. A similar theorem
statement from [5] also holds between radix expansions in base $-2+i$ and its
state graph. It can be found in the appendix (see Theorem B.1). The
descriptions that follow pertain to Figure 1.

If a complex number has a unique radix expansion in base $-n+i$, $n\geq 3$,
then $p=q=r$ and this triple is perpetually in the state $(0,0,0)$. Complex
numbers with precisely two distinct radix expansions correspond to paths that
eventually exit the initial state $(0,0,0)$ but remain in the bolded red
subgraph that does not distinguish between $p$ and $q$. Complex numbers with
three distinct radix expansions eventually exit the initial state $(0,0,0)$
and ultimately are trapped in one of the two loops of period three at the
bottom of the diagram.

We provide an example to illustrate how to read the graph.

######  Example 2.6.

The complex number $\frac{-23-10i}{17}$ has the following three radix
expansions in base $b=-3+i$:

$\begin{split}p&=(0;\overline{4,0,9,}),\\\ q&=(1;\overline{9,4,0,}),\\\
r&=(1,5,5;\overline{0,9,4,}).\\\ \end{split}$

The bar over the digits to the right of the radix point indicates a repetition
of those digits with period three. The path that this number corresponds to in
the state graph is the path that moves along the states

pqrpqrpqrrpqpqrqrp

.

This path also captures the complex number
$\frac{-108+24i}{17}=21.\overline{409}=22.\overline{904}=176.\overline{094}$.
The distances between pairs of coefficients of the same power of $b$ is the
same as those in the previous triple of expansions.

A list of observations about radix expansions are made from the state graph in
[5]. We state an additional observation.

######  Corollary 2.7.

Suppose $x$ and $y$ are two distinct radix expansions of the same complex
number in base $-n+i$ where $n\geq 2$. Let $k\in\mathbb{Z}$ be the first index
at which a pair of digits $x_{k}$ and $y_{k}$ are not equal. Then
$x_{k}-y_{k}=\pm 1$.

###### Proof.

The analysis that follows corresponds to the graph in Figure 1 governing radix
expansions in base $-n+i$ for base $n\geq 3$. A similar analysis can be made
of the graph governing the case $n=2$ in Appendix B which results in the same
conclusion.

If $x$ and $y$ are the only distinct radix expansions of the complex number
they represent, then they correspond to a path that, eventually, leaves the
initial state $(0,0,0)$ and then remains in the bolded red subgraph of Figure
1. Without loss of generality, we label $p=q=x$ and $r=y$. The first instance
that an entry of $r$ differs from that of $p$ is when the path leaves the
state $(0,0,0)$. From the graph, we see that the pair of digits between $r$
and $p$ differ by $\pm 1$ at that index of the radix expansions.

If $x$ and $y$ are two of three distinct radix expansions, then the path they
correspond to ultimately enters, and is trapped, in one of the two loops of
period three at the bottom of the diagram. If either $x$ or $y$ fit the role
of $r$, then the expansions again differ for the first time when they leave
state $(0,0,0)$. If neither $x$ or $y$ can be assigned the role of $r$, then
the two expansions differ at a change of state that enters one of the two
loops of period three. There are four of these edges and they all indicate
that the digits of $p$ and $q$ differ by $\pm 1$. ∎

Let us now return to the context of fractal geometry and see what this
observation can afford us.

######  Definition 2.8.

Let $b=-n+i$ where $n\geq 2$. We say that a subset $D\subset\Lambda_{b}$ is
separated if for $d,d^{{}^{\prime}}\in D$, we have $|d-d^{{}^{\prime}}|>1$
whenever $d\neq d^{{}^{\prime}}$.

######  Lemma 2.9.

Let $b=-n+i$ where $n\geq 2$. Suppose $D\subset\Lambda_{b}$ is separated.

Every element of $C_{D}$ has a unique radix expansion that only uses digits in
$D$.

###### Proof.

Suppose $z\in C_{D}$. By definition $z$ has a radix expansion $q$ that only
uses digits in $D$. To argue for uniqueness, we observe by corollary 2.7 that
any other radix expansion of $z$, if one exists, must use a digit that differs
by $\pm 1$ from a digit in $q$. By assumption, this digit must not be in $D$.
It follows that $q$ is unique. ∎

This observation allows us to define a map that can play the role of $x\mapsto
rx\mod{1}$ in our setting.

######  Definition 2.10.

Let $b=-n+i$ where $n\geq 2$. Suppose $D\subset\Lambda_{b}$ is separated. Let
$T_{b}$ be the map

$T_{b}:C_{D}\rightarrow C_{D}$
$z=\sum_{k=1}^{\infty}d_{k}b^{-k}\mapsto\sum_{k=1}^{\infty}d_{k+1}b^{-k}=w.$

where $d_{k}\in D$ for all $k$.

A nonempty closed subset $Y\subset C_{D}$ is $\times b$-invariant if
$T_{b}(Y)\subset Y$. We say that $Y\subset S$ is multiplicatively invariant if
it is $\times b$-invariant for some $b=-n+i$ where $n\geq 2$.

######  Example 2.11.

The restricted digit Cantor set $C_{D}$ is $\times b$-invariant if the digit
set $D$ is separated.

It is natural to ask what results for multiplicatively invariant subsets of
$[0,1]$ can be replicated for their analogues in the complex plane. We recall
the main objective of this document.

######  Theorem 2.12.

Let $b=-n+i$ where $n\geq 3$ and assume $D\subset\Lambda_{b}$ is separated.
Let $g:D^{\mathbb{N}}\rightarrow C_{D}$ be the map $(d_{j})_{j\geq
1}\mapsto\sum_{j\geq 1}d_{j}b^{-j}$. If $A\subset D^{\mathbb{N}}$ is a
subshift, then

  * (i)

$g(A)=\\{\sum_{k=1}^{\infty}a_{k}b^{-k}:(a_{k})_{k\geq 1}\in A\\}$ is $\times
b$-invariant,

  * (ii)

$\dim_{B}g(A)=\frac{\mathcal{E}(A)}{\log{|b|}}.$

  * (iii)

If $Y\subset C_{D}$ is a $\times b$-invariant set, then $\dim_{H}Y=\dim_{B}Y.$

To prove these claims we adopt the approach of [10] and work with shifts of
scalings of $S:=C_{\Lambda_{b}}$.

######  Definition 2.13.

Let $b=-n+i$ where $n$ is a positive integer and $D\subset\Lambda_{b}$. We
call a set of the form

(13) $0.d_{1}d_{2}\cdots d_{m}+b^{-m}S:=\bigg{\\{}z=\sum_{k\geq
1}z_{k}b^{-k}\in\mathbb{C}:z_{k}=d_{k}\in
D\;\text{for}\;k=1,2,\ldots,m\bigg{\\}}$

an $m$-tile with digits in $D$. Whenever the set $D$ is unspecified, we mean
$D=\Lambda_{b}$.

One application of these tiles is to use them to compute the box-counting
dimension of a base-$b$ restricted digit set $C_{D}$. The following result
from [10] formulates the box-counting dimension of a subset of $S$ in terms of
the number of $m$-tiles required to cover it.

######  Lemma 2.14 (S. Pedersen, V. Shaw, lemma 5.2).

Let Y be a nonempty subset of $S$. For a fixed integer $m\geq 1$, let
$N_{m}(Y)$ denote the smallest number of $m$-tiles needed to cover Y. Then the
box-counting dimension of Y exists if and only if
$\lim_{m\rightarrow\infty}\frac{\log{N_{m}(Y)}}{m\log{|b|}}$ exists, and, if
so, this limit is the box-counting dimension of Y.

Let us apply this tool to the set $C_{D}$.

######  Theorem 2.15.

Let $b=-n+i$ where $n\geq 2$ and suppose $D$ is a subset of $\Lambda_{b}$.
Then

(14) $\dim_{B}C_{D}=\frac{\log{|D|}}{\log{|b|}}.$

###### Proof.

There are $|D|^{m}$ words of length $m$ that use digits in $D$. If we index
over all such words we have

(15) $C_{D}\subset\cup_{(d_{1},d_{2},\ldots,d_{m})}0.d_{1}d_{2}\cdots
d_{m}+b^{-m}S$

Therefore $N_{m}(C_{D})\leq|D|^{m}$. On the other hand, we can show that every
one of the $|D|^{m}$ tiles in the union contains a point in $C_{D}$ that is
not also contained in any of the other $m$-tile. Namely, the tile
$0.d_{1}d_{2}\cdots d_{m}+b^{-m}S$ contains the complex number
$0.d_{1}d_{2}\cdots d_{m}$. If this number were in another $m$-tile, it would
have more than one radix expansion. This is impossible because no expansions
that correspond to paths with distinct expansions in Figure 1 has a tail of
zeros. This can be verified by direct inspection. The same inspection can be
made of the graph governing the case when $b=-2+i$.

It follows that every one of the $|D|^{m}$ tiles is necessary to cover $C_{D}$
and we have that $N_{m}(C_{D})\geq|D|^{m}$. Combining both inequalities allows
us to conclude that

(16)
$\lim_{m\to\infty}\frac{\log{N_{m}(C_{D})}}{\log{|b|}}=\frac{\log{|D|}}{\log{|b|}}.$

We obtain our result by appealing to Lemma 2.14. ∎

######  Remark 2.16.

This formula already exists in the literature, see [10]. What is different is
the presence of a separation condition. The original statement assumes that
the digits are separated by a distance greater than $n$, where $b=-n+i$. We
have strengthened the result by eliminating the separation condition.

In general, $m$-tiles are not disjoint since radix expansions are not unique.
When the separation condition is enforced, we can claim this.

######  Lemma 2.17.

Let $b=-n+i$ where $n\geq 2$ and assume $D\subset\Lambda_{b}$ is separated.
For a fixed positive integer $m$, any two distinct $m$-tiles with digits in
$D$ are disjoint.

###### Proof.

If the intersection is nonempty then there exists a complex number with at
least two radix expansions with digits in $D$. It follows that the distance
between the pair of digits at which they first differ is greater than $1$.
This contradicts Corollary 2.7. ∎

It is convenient to recognize that, under the separation condition, the
topology of $C_{D}$ is generated by cylinder sets.

######  Lemma 2.18.

Let $b=-n+i$ where $n\geq 2$ and assume $D\subset\Lambda_{b}$ is separated.

The collection of sets of the form $0.d_{1}d_{2}\cdots d_{m}+b^{-m}C_{D}$,
where $m$ is some positive integer and $d_{j}\in D$ for $j=1,2,\ldots,m$, are
a basis for the topology on $C_{D}$.

###### Proof.

Let us denote the proposed basis by $\mathcal{U}$. We first must show that
$\mathcal{U}$ is the basis of some topology.

We require that any element of $C_{D}$ is contained in some element of
$\mathcal{U}$. By the definition of $C_{D}$, every $z\in C_{D}$ is an element
of $0.z_{1}z_{2}\cdots z_{m}+b^{-m}C_{D}$ where $z_{j}\in D$ for every $m$ and
thus the requirement is met. We also require that whenever an element of
$C_{D}$ is contained in the intersection of two sets in $\mathcal{U}$, it is
contained in a set in $\mathcal{U}$ that is a subset of the intersection. By
the Lemma 2.17, the separation condition implies that two sets of the form
$0.d_{1}d_{2}\cdots d_{m}+b^{-m}S$ and
$0.d_{1}^{{}^{\prime}}d_{2}^{{}^{\prime}}\cdots d_{n}^{{}^{\prime}}+b^{-n}S$
are disjoint whenever $d_{k}\neq d_{k}^{{}^{\prime}}$ for some
$k\in\\{1,2,\ldots,\min(m,n)\\}$. Otherwise, one is a subset of the other.
Since $C_{D}\subset S$, the collection of sets under consideration also have
this property. It follows that the second requirement is met. This verifies
that $\mathcal{U}$ is a basis for a topology on $C_{D}$.

The statement that $\mathcal{U}$ generates the topology on $C_{D}$ inherited
from $\mathbb{C}$ can be restated as
$\mathcal{T}_{\mathcal{U}}=\mathcal{T}_{\mathcal{B}}$ where
$\mathcal{T}_{\mathcal{U}}$ is the topology generated by $\mathcal{U}$ and
$\mathcal{T}_{\mathcal{B}}$ is the topology generated by balls intersected
with $C_{D}$. To show this we argue that balls intersected with $C_{D}$ are
elements of $\mathcal{T}_{\mathcal{U}}$ and then show that elements of
$\mathcal{U}$ are in $\mathcal{T}_{\mathcal{B}}$.

Suppose $z\in C_{D}\cap B(z_{0},r)$ where $B(z_{0},r)$ is a ball centered at
$z_{0}$ with radius $r>0$. Since $z\in C_{D}$, we know it has a radix
expansion of the form $(0.z_{1}z_{2}\cdots)$ where $z_{j}\in D$. We claim that
if $M$ is chosen sufficiently large that

(17) $n^{2}(|b|^{M}(|b|-1))^{-1}<r-|z-z_{0}|,$

then the set $0.z_{1}z_{2}\cdots z_{M}+b^{-M}C_{D}$, which contains $z$, is a
subset of $C_{D}\cap B(z_{0},r)$. To see this, suppose $w\in
0.z_{1}z_{2}\cdots z_{M}+b^{-M}C_{D}$ and observe that

(18) $\begin{split}|z-w|&\leq\sum_{j=M+1}^{\infty}|z_{j}-w_{j}||b|^{-j}\\\
&\leq\sum_{j=0}^{\infty}n^{2}|b|^{-(m+1)}|b|^{-j}\\\
&=n^{2}(|b|^{M}(|b|-1))^{-1}.\\\ \end{split}$

It follows that

(19) $0.z_{1}z_{2}\cdots z_{M}+b^{-M}C_{D}\subset B(z,r-|z-z_{0}|)\subset
B(z_{0},r).$

This shows that $\mathcal{T}_{\mathcal{B}}\subset\mathcal{T}_{\mathcal{U}}$.

To obtain the converse, observe that for any fixed $m$, the union of all the
sets of the form $0.d_{1}d_{2}\cdots d_{m}+b^{-m}C_{D}$ is equal to $C_{D}$.
Therefore any one of them can be expressed as the complement of a finite union
of sets which are closed in $(C_{D},\mathcal{T}_{\mathcal{B}})$. This gives us
$\mathcal{T}_{\mathcal{B}}\supset\mathcal{T}_{\mathcal{U}}$. This concludes
the proof. ∎

######  Lemma 2.19.

Let $b=-n+i$ where $n\geq 2$ and suppose $D\subset\Lambda_{b}$ is separated.
The map

$g:D^{\mathbb{N}}\rightarrow C_{D}$ $(d_{j})^{j\geq 1}\mapsto(d_{j})_{j\geq
1}\mapsto\sum_{j=1}^{\infty}d_{j}b^{-j}$

is a topological conjugacy between the subshift $(D^{\mathbb{N}},\sigma)$ and
the system $(C_{D},T_{b})$. That is, the map $g$ is a homeomorphism and
$T_{b}\circ g=g\circ\sigma$.

###### Proof.

The bijective correspondence between a sequence of digits and the members of
$C_{D}$ follows from two facts. Firstly, the elements of $C_{D}$ have radix
expansions determined by a sequence of digits in $D$ by definition. Secondly,
the digits of the expansion of a given complex number that uses digits in $D$
is unique by Lemma 2.9. The continuity of $g$ and its inverse follows from
observing the bijective correspondence between cylinder sets in
$D^{\mathbb{N}}$ and the sets of the form $0.d_{1}d_{2}\cdots
d_{m}+b^{-m}C_{D}$ and then invoking Lemma 2.18. To see that $g$ intertwines
the dynamics, observe that

(20) $(T_{b}\circ g)((d_{j})_{j\geq
1})=T_{b}\big{(}\sum_{j=1}^{\infty}d_{j}b^{-j}\big{)}=\sum_{j=1}^{\infty}d_{j+1}b^{-j}=g((d_{j+1})_{j\geq
1})=(g\circ\sigma)((d_{j})_{j\geq 1})$

∎

######  Corollary 2.20.

Let $b=-n+i$ where $n\geq 3$ and assume $D\subset\Lambda_{b}$ is separated.
The set $C_{D}\subset\mathbb{C}$ is a Cantor space, that is, a totally
disconnected compact metric space with no isolated points.

###### Proof.

This is an immediate consequence of Lemma 2.19 since it is well known that
sequence spaces equipped with the product topology are Cantor spaces. ∎

When we compute the box-counting dimension of subsets of a multiplicatively
invariant set, we are examining not only a subset of $S$ but a subset of some
$C_{D}$. To this end, we need only count $m$-tiles specifying digits from a
sufficiently separated digit set $D$. The following result states that we can
still capture the box-counting dimension of a non-empty subset of $S_{0}$ with
$m$-tiles that only use certain digits as long as we can cover the subset
using such $m$-tiles.

######  Lemma 2.21.

Let $b=-n+i$ where $n\geq 3$ and $D\subset\Lambda_{b}$. Assume Y is a nonempty
subset of $C_{D}$. For a fixed integer $m\geq 1$, let $\tilde{N_{m}}(Y)$
denote the smallest number of $m$-tiles with digits in $D$ needed to cover
$Y$. Then the box-counting dimension of $Y$ exists if and only if
$\lim_{m\rightarrow\infty}\frac{\log{\tilde{N_{m}}(Y)}}{m\log{|b|}}$ exists,
and, if so, this limit is the box-counting dimension of Y.

###### Proof.

First note that the set of covers of $Y$ by $m$-tiles with digits in $D$ is
nonempty because $C_{D}$ is contained in the union over all such tiles.

It follows immediately from the definitions that
$N_{m}(Y)\leq\tilde{N}_{m}(Y)$. On the other hand, suppose $T$ is an $m$-tile
that uses digits in $\Lambda_{b}\setminus D$. If $T$ intersects $Y$ then the
elements of the intersection have at least $2$ distinct radix expansions.

From Lemma 2.4, we can deduce that if $S$ intersects a tile of the form $S+g$
where $g\in\mathbb{Z}[i]$, then $g$ is one of, at most, eight nonzero values.
Since $T$ (or any $m$-tile) is a translate of $S$ scaled and rotated by
$b^{-m}$, it follows that the number of $m$-tiles that intersect $T$ is
bounded by eight.

We conclude that

(21) $N_{m}(Y)\leq\tilde{N}_{m}(Y)\leq 8N_{m}(Y)$

and, equivalently,

(22) $\frac{1}{8}\tilde{N}_{m}(Y)\leq N_{m}(Y)\leq\tilde{N}_{m}(Y).$

Taking logarithms and limits yields the result. ∎

For the remainder of this section, we will be only consider the box-counting
dimension of multiplicatively invariant subsets and thus use the function
$N_{m}$ to denote the version that only counts $m$-tiles with digits in some
$D$. We can now prove $(i)$ and $(ii)$ of Theorem 2.12: the relationship
between multiplicatively invariant subsets and subshifts. We will use the
notation $[a_{1},a_{2},\ldots,a_{n}]:=\\{(x_{k})_{k\geq
1}:x_{k}=a_{k}\;\text{for}\;k=1,2,\ldots,n\\}$ to denote cylinder sets in the
shift space.

######  Theorem 2.22.

Let $b=-n+i$ where $n\geq 3$ and assume $D\subset\Lambda_{b}$ is separated.
Let $g:D^{\mathbb{N}}\rightarrow C_{D}$ be the map $(d_{j})_{j\geq
1}\mapsto\sum_{j\geq 1}d_{j}b^{-j}$. If $A\subset D^{\mathbb{N}}$ is a
subshift, then

  * (i)

$g(A)=\\{\sum_{k=1}^{\infty}a_{k}b^{-k}:(a_{k})_{k\geq 1}\in A\\}$ is $\times
b$-invariant,

  * (ii)

$\dim_{B}g(A)=\frac{\mathcal{E}(A)}{\log{|b|}}.$

Moreover, if $Y\subset C_{D}$ is $\times b$-invariant, then $g^{-1}(Y)$ is a
subshift satisfying $\mathcal{E}(g^{-1}(Y))/\log{|b|}=\dim_{B}Y$.

###### Proof.

By Lemma 2.19, we have that $g(A)$ is both closed and $T_{b}$-invariant since
$A$ is closed and shift invariant.

We now argue that the box-counting dimension of $g(A)$ is equal to the
topological entropy of $A$ normalized by $\log{|b|}$.

By Lemma 2.21, the box-counting dimension of $g(A)$ can be computed using the
function $N_{m}$ which counts the smallest number of $m$-tiles using digits in
$D$ needed to cover $g(A)$. By way of the map $g$, there is a bijective
correspondence between those tiles and the cylinder sets
$[d_{1},d_{2},\ldots,d_{m}]$. Therefore there is a bijective correspondence
between the $m$-tiles using digits in $D$ and the words of length $m$ that can
be written using the digits in $D$. Using the notation developed for subshifts
in the previous section, we have

(23) $N_{m}(g(A))=|\mathcal{L}_{m}(A)|$

and in particular,

(24)
$\frac{\log{N_{m}(g(A))}}{m\log{|b|}}=\frac{\log{|\mathcal{L}_{m}(A)|}}{m\log{|b|}}.$

Taking limits as $m\rightarrow\infty$ yields the result.

Now we suppose we start with a $\times b$-invariant subset $Y$. We again
invoke the conjugacy properties of $g$ to claim that $g^{-1}(Y)$ is closed and
shift invariant. The relationship between the topological entropy of
$g^{-1}(Y)$ and the box-counting dimension of $Y$ follows from the preceding
argument. ∎

We end by showing that the box-counting and Hausdorff dimensions of a
multiplicatively invariant subset of the complex plane are equal. We do this
by emulating an argument of Furstenberg’s from [3].

To do this, however, we require an alternative method of capturing the
Hausdorff dimension. The idea is to restrict the $\delta$-covers used to
define the Hausdorff measure by only using $m$-tiles. We begin with three
technical lemmas.

######  Lemma 2.23.

Let $b=-n+i$ where $n\geq 3$. Fix a positive integer $m$. There exists a
bound, independent of $m$, on the number of $m$-tiles that any ball with
diameter less than or equal to $|b|^{-m}\operatorname{diam}{S}$ intersects.

###### Proof.

Observe that the diameter of an $m$-tile $0.d_{1}d_{2}\cdots d_{m}+b^{-m}S$ is
$|b|^{-m}\operatorname{diam}{S}$.

Firstly, there exists a finite number of sets of the form $g+S$, where
$g\in\mathbb{Z}[i]$, that any ball of fixed diameter $\delta$ can intersect.
This is because the set $S$ is bounded and thus we can bound the modulus of
any Gaussian integer $g$ that satisfies the inequality $|w-(g+z)|<\delta/2$
where $w$ is the center of the ball and $z$ is some element of $S$. Namely,
the reverse triangle inequality yields $|g|<\delta/2+|w-z|$.

Let $M$ be the maximum number of translates of $S$ by Gaussian integers that a
ball of diameter $\operatorname{diam}{S}$ can intersect. If a ball with
diameter less than or equal to $|b|^{-m}\operatorname{diam}{S}$ intersects
more than $M$ $m$-tiles, then we can scale all the $m$-tiles and the ball by
$b^{m}$ to obtain a ball of diameter less than or equal to
$\operatorname{diam}{S}$ that intersects more than $M$ translates of $S$.

It follows that $M$ is the bound we want. ∎

######  Lemma 2.24.

Let $Y$ be a subset of $S$ and fix $s\geq 0$. For any $\delta>0$, let

(25)
$\mathcal{T}_{\delta}^{s}(Y):=\inf{\bigg{\\{}\sum_{k=1}^{\infty}(\operatorname{diam}{T_{k}})^{s}:\\{T_{k}\\}\;\text{is
a $\delta$-cover of}\;Y\;\text{where each $T_{k}$ is an
$m_{k}$-tile}\bigg{\\}}}.$

Then $\dim_{H}{Y}=\inf{\\{s\geq 0:\mathcal{T}^{s}(Y)=0\\}}$ where
$\mathcal{T}^{s}(Y):=\lim_{\delta\rightarrow
0^{+}}\mathcal{T}_{\delta}^{s}(Y)$.

###### Proof.

Suppose that $\\{B_{k}\\}$ is a $\delta$-cover of $Y$ by balls. Since we
ultimately are concerned with the limit as $\delta$ tends to zero, we may
assume $\delta\in(0,1)$. For each $k$, we can find an integer $m_{k}$ such
that
$|b|^{-(m_{k}+1)}\operatorname{diam}{S}<\operatorname{diam}{B_{k}}\leq|b|^{-m_{k}}\operatorname{diam}{S}$.
Each ball $B_{k}$ intersects some finite number of $m_{k}$-tiles that also
intersect $Y$. The collection of these $m_{k}$-tiles, over all $k$, form a
$|b|\delta$-cover of $Y$. Let $T^{(k)}_{j}$ denote the $j$th $m_{k}$-tile that
intersects $B_{k}$. Suppose $s\geq 0$. Let $M$ be an upper bound on the number
of $m_{k}$-tiles that $B_{k}$ can intersect. This bound exists by Lemma 2.23.
It follows that

(26) $\displaystyle\sum_{k}\sum_{j}(\operatorname{diam}{T^{(k)}_{j}})^{s}$
$\displaystyle\leq\sum_{k}M(\operatorname{diam}{T^{(k)}_{1}})^{s}$ (27)
$\displaystyle=M\sum_{k}(|b|^{-m_{k}}\operatorname{diam}{S})^{s}$ (28)
$\displaystyle=M|b|^{s}\sum_{k}(|b|^{-(m_{k}+1)}\operatorname{diam}{S})^{s}$
(29) $\displaystyle\leq M|b|^{s}\sum_{k}(\operatorname{diam}{B_{k}})^{s}.$

Since $\\{T^{(k)}_{j}\\}$ is a collection of $m_{k}$-tiles that form a
$|b|\delta$-cover of $Y$, we obtain

(30) $\mathcal{T}_{|b|\delta}^{s}(Y)\leq
M|b|^{s}\sum_{k}(\operatorname{diam}{B_{k}})^{s}.$

Since the $\delta$-cover of balls is arbitrary, this implies
$\mathcal{T}_{|b|\delta}^{s}(Y)\leq M|b|^{s}\mathcal{B}_{\delta}^{s}(Y)$ (see
Proposition 1.5). The Hausdorff measure is defined using arbitrary countable
$\delta$-covers and so we immediately have
$\mathcal{H}_{|b|\delta}^{s}(Y)\leq\mathcal{T}_{|b|\delta}^{s}(Y)$. Taking the
limits as $\delta\rightarrow 0^{+}$ yields

(31) $\mathcal{H}^{s}(Y)\leq\mathcal{T}^{s}(Y)\leq
M|b|^{s}\mathcal{B}^{s}(Y).$

From [2] it is known that $\mathcal{H}^{s}(Y)$ and $\mathcal{B}^{s}(Y)$ both
have the property that they are $+\infty$ for $s<\dim_{H}Y$ and $0$ for
$s>\dim_{H}Y$. It follows that $\mathcal{T}^{s}(Y)$ shares this property.
Therefore

(32) $\inf{\\{s\geq 0:\mathcal{T}^{s}(Y)=0\\}}=\inf{\\{s\geq
0:\mathcal{H}^{s}(Y)=0\\}}=\dim_{H}Y.$

∎

We can also restrict $m_{k}$-tiles in Lemma 2.24 to those containing tiles
with digits in $D$, provided that we can cover $Y$ with such a collection of
sets.

######  Lemma 2.25.

Let $Y\subset C_{D}$. For any $\delta>0$, let
$\mathcal{\hat{T}}_{\delta}^{s}(Y)$ be the modification of
$\mathcal{T}_{\delta}^{s}(Y)$ in (25) where the infimum is taken over all
$\delta$-covers of $m_{k}$-tiles of $Y$ with digits in $D$. Then
$\dim_{H}{Y}=\inf{\\{s\geq 0:\mathcal{\hat{T}}^{s}(Y)=0\\}}$ where
$\mathcal{\hat{T}}^{s}(Y):=\lim_{\delta\rightarrow
0^{+}}\mathcal{\hat{T}}_{\delta}^{s}(Y)$.

###### Proof.

The assumption $Y\subset C_{D}$ ensures that such covers exist. The tiles of
these covers are a subset of all $m$-tiles. Therefore the bound on the number
of $m$-tiles with digits in $D$ that intersect a ball of diameter
$|b|^{-m}\operatorname{diam}{S}$ is also bounded by the number referenced in
Lemma 2.23. The existence of this bound, which is independent of $m$, means we
can repeat the argument in the proof of Lemma 2.24 to obtain the result. ∎

We now prove $(iii)$ of theorem 2.12. This is a direct application of
Furstenberg’s proof technique from [3].

######  Theorem 2.26.

Let $Y\subset C_{D}$ be a $\times b$-invariant set. Then
$\dim_{H}Y=\dim_{B}Y.$

###### Proof.

It is known that $\dim_{H}E\leq\dim_{B}E$ for any $E\subset\mathbb{R}^{n}$
whenever the box-counting dimension exists, see [2]. We need to show that
$\dim_{B}Y\leq\dim_{H}Y$. Furthermore, we can assume that $\dim_{B}Y>0$ since
otherwise both dimensions are zero and there is nothing more to show. The
strategy is to argue that $s\leq\dim_{H}Y$ whenever $0\leq s<\dim_{B}Y$.

By definition, we have $\mathcal{T}^{s}(Y)=0$ if $s>\dim_{H}Y$. Therefore, to
arrive at $s\leq\dim_{H}Y$, it suffices to show that $\mathcal{T}^{s}(Y)>0$.
The latter will hold if there exists $c>0$ such that every $\delta$-cover of
$Y$ by $m_{k}$-tiles $\\{T_{k}\\}$ using digits in $D$ satisfies $\sum_{k\geq
1}(\operatorname{diam}{T_{k}})^{s}\geq c>0$. For a given $s$, we show this for
$c=(\operatorname{diam}{S})^{s}$.

Since $Y$ is a closed subset of the compact space $C_{D}$, it is compact.
Thus, by Lemma 2.18, we need only consider finite covers of $m_{k}$-tiles with
digits in $D$. Therefore what we want to show is: if
$Y\subset\cup_{k=1}^{K}T_{k}$ and $s<\dim_{B}Y$, where $\\{T_{k}\\}$ is a
finite collection of $m_{k}$-tiles, then $\sum_{k=1}^{K}|b|^{-sm_{k}}\geq 1$.
Note that we have divided out by $(\operatorname{diam}{S})^{s}$.

To proceed, we rephrase this statement by identifying elements of $Y$ with
their sequences of digits in $D$.

Let $\hat{Y}=\\{(y_{k})_{k\geq 1}\in D^{\mathbb{N}}:\sum_{k\geq
1}y_{k}b^{-k}\in Y\\}$. By Theorem 2.22, this is a subshift with the left-
shift operator. Let $L=\cup_{n\geq 1}\Lambda_{b}^{n}$. This is the set of all
$n$-tuples of elements of $\Lambda_{b}$. Let $R$ be the subset of $L$
containing those tuples which occur as finite subwords of sequences in
$\hat{Y}$. The set $\mathcal{L}_{n}(\hat{Y})$ can then be viewed as the
elements of $R$ of length $n$. The set $R$ is a semigroup with respect to
concatenation. Let us say that a word $\rho$ divides a word
$\rho^{{}^{\prime}}$ if $\rho^{{}^{\prime}}=\rho\rho_{1}$ for some
$\rho_{1}\in L$.

By Lemma 2.19, there is a bijective correspondence between $m_{k}$-tiles
$T_{k}=0.d_{1}^{(k)}d_{2}^{(k)}\cdots d_{m_{k}}^{(k)}+b^{-m_{k}}S$ with digits
in $D$ and cylinder sets
$[d_{1}^{(k)},d_{2}^{(k)},\ldots,d_{m_{k}}^{(k)}]\subset D^{\mathbb{N}}$. The
latter has a bijective correspondence with words of the form
$d_{1}^{(k)}d_{2}^{(k)}\cdots d_{m_{k}}^{(k)}$ in $R$. The containment
$Y\subset\cup_{k=1}^{K}T_{k}$ holds if and only if $\hat{Y}$ is contained in
$\cup_{k=1}^{K}[d_{1}^{(k)},d_{2}^{(k)},\ldots,d_{m_{k}}^{(k)}]$. This is
equivalent to the statement that for any $\rho\in R$ of sufficient length,
there exists $k=1,2,\ldots,K$ such that $\rho_{k}=d_{1}^{(k)}d_{2}^{(k)}\ldots
d_{m_{k}}^{(k)}$ divides $\rho$.

Let $N$ denote the length threshold for division by an element of
$\\{\rho_{k}\\}_{k=1}^{K}$. Recall that we want to show that if
$Y\subset\cup_{k=1}^{K}T_{k}$ and $s<\dim_{B}Y$, then
$\sum_{k=1}^{K}|b|^{-sm_{k}}\geq 1$. We can rephrase this implication using
the equivalences highlighted in the previous paragraph: if there exists a
finite collection $\\{\rho_{k}\\}_{k=1}^{K}\subset R$ such that whenever
$\rho\in R$ is of sufficient length, there exists $k=1,2,\ldots,K$ such that
$\rho_{k}$ divides $\rho$ and $s<\dim_{B}Y$, then
$\sum_{k=1}^{K}|b|^{-sl(\rho_{k})}\geq 1$ where $l(\rho_{k})$ is the length of
the word $\rho_{k}$.

Let us now prove this implication.

We proceed by contradiction and assume that
$\sum_{k=1}^{K}|b|^{-sl(\rho_{k})}<1$. Let $\langle\rho_{k}\rangle$ be the
semigroup generated by $\\{\rho_{k}\\}_{k=1}^{K}$ using concatenation. We have

(33)
$\begin{split}\sum_{\langle\rho_{k}\rangle}|b|^{-sl(\rho_{k_{1}}\rho_{k_{2}}\ldots\rho_{k_{n}})}&=\sum_{n=1}^{\infty}\sum_{(k_{1},k_{2},\ldots,k_{n})}|b|^{-sl(\rho_{k_{1}}\rho_{k_{2}}\ldots\rho_{k_{n}})}\\\
&=\sum_{n=1}^{\infty}\bigg{(}\sum_{(k_{1},k_{2},\ldots,k_{n})}\prod_{i=1}^{n}|b|^{-sl(\rho_{k_{i}})}\bigg{)}\\\
&=\sum_{n=1}^{\infty}\bigg{(}\sum_{k=1}^{K}|b|^{-sl(\rho_{k})}\bigg{)}^{n}.\\\
\end{split}$

The last sum is a convergent geometric series by our assumption that
$\sum_{k=1}^{K}|b|^{-sl(\rho_{k})}<1$. We can use this to prove that
$\sum_{R}|b|^{-sl(\rho)}$ converges. By the shift invariance of $\hat{Y}$, we
have that whenever $\rho_{1}\rho_{2}\in R$, it must be that
$\rho_{1},\rho_{2}\in R$. By assumption, the set $\\{\rho_{k}\\}_{k=1}^{K}$
has the property that every element of $R$ of length at least $N$ is divisible
by one of the elements of $\\{\rho_{k}\\}_{k=1}^{K}$. Combining these two
properties allows us to divide until there is no more room to do so. This
yields

(34) $\rho=\rho_{k_{1}}\rho_{k_{2}}\ldots\rho_{k_{n}}\rho_{j}^{{}^{\prime}}$

where $\rho_{j}^{{}^{\prime}}$ is some element of $R$ that is of insufficient
length. The set of these remainders is finite since there are only finitely
many words whose length is less than $N$, say $J$ of them. It suffices to
argue that $\sum_{R}|b|^{-sl(\rho)}$ is finite when we restrict the index set
to those words of length at least $N$. Observe that

(35) $\displaystyle\sum_{\rho\in R,l(\rho)\geq N}|b|^{-sl(\rho)}$
$\displaystyle=\sum_{\rho\in R,l(\rho)\geq
N}|b|^{-sl(\rho_{k_{1}}\rho_{k_{2}}\ldots\rho_{k_{n}}\rho_{j}^{{}^{\prime}})}$
(36)
$\displaystyle<\sum_{\langle\rho_{k}\rangle}\sum_{j=1}^{J}|b|^{-sl(\rho_{k_{1}}\rho_{k_{2}}\ldots\rho_{k_{n}}\rho_{j}^{{}^{\prime}})}$
(37)
$\displaystyle<J\sum_{\langle\rho_{k}\rangle}|b|^{-sl(\rho_{k_{1}}\rho_{k_{2}}\ldots\rho_{k_{n}})}.$

The last quantity is finite since the quantity in (33) is finite. This implies
that $\sum_{R}|b|^{-sl(\rho)}$ converges. On the other hand, we can show that
this same infinite series diverges because $s<\dim_{B}Y$.

By Theorem 2.22, the box-counting dimension of $Y$ is equal to
$\frac{\mathcal{E}(\hat{Y})}{\log{|b|}}$ where $\mathcal{E}(\hat{Y})$ is the
topological entropy of the subshift $\hat{Y}$. The inequality $s<\dim_{B}Y$
implies that, for all sufficiently large $m$, we have
$s<\frac{\log|\mathcal{L}_{m}(\hat{Y})|}{m\log{|b|}}$. Therefore
$|\mathcal{L}_{m}(\hat{Y})||b|^{-sm}>1$ for all sufficiently large $m$.

If we enumerate over $R$ in increasing length, then we have
$\sum_{R}|b|^{-sl(\rho)}=\sum_{m=1}^{\infty}|\mathcal{L}_{m}(\hat{Y})||b|^{-sm}$.
The latter series diverges to $+\infty$. We have established our
contradiction. If $s<\dim_{B}Y$, then we must have
$\sum_{k=1}^{K}|b|^{-sl(\rho_{k})}\geq 1$. This means that $s\leq\dim_{H}Y$
for all $s<\dim_{B}Y$. It follows that $\dim_{B}Y\leq\dim_{H}Y$. We recall
that the reverse inequality always holds whenever the box-counting dimension
exists. Therefore we achieve equality. ∎

We end this section with the observation that we can now express the Hausdorff
dimension of a base-$b$ restricted digit Cantor set with a separated digit set
$D$ in terms of the cardinality of $D$ and the modulus of $b$.

######  Corollary 2.27.

Let $b=-n+i$ where $n\geq 2$ and suppose $D\subset\Lambda_{b}$ is separated.
Then

(38) $\dim_{H}C_{D}=\frac{\log{|D|}}{\log{|b|}}.$

###### Proof.

When $D$ is separated, the set $C_{D}$ is a $\times b$-invariant subset of
itself. Direct applications of Theorem 2.26 and Theorem 2.15 yield the result.
∎

## References

  * [1] T. Austin.  A new dynamical proof of the Shmerkin-Wu theorem.  Journal of Modern Dynamics, 18:1–11, 2022.
  * [2] K. Falconer.  Fractal Geometry:Mathematical Foundations and Applications.  John Wiley and Sons, 1990.
  * [3] H. Furstenberg.  Disjointedness in ergodic theory, minimal sets, and a problem in Diophantine approximation.  Math. Systems Theory, 1:1–49, 1967.
  * [4] H. Furstenberg.  Intersections of Cantor Sets and Transversality of Semigroups, pages 41–59 in Problems in Analysis.  Princeton University Press, 1970.
  * [5] W. J. Gilbert.  Complex numbers with three radix expansions.  Can. J. Math., 34:1335–1348, 1982.
  * [6] D. Glasscock, J. Moreira, and F. Richter.  Additive and geometric transversality of fractal sets in the integers.  arXiv preprint arXiv:2007.05480, 2021.
  * [7] M. Hochman and P. Shmerkin.  Local entropy averages and projections of fractal measures.  Ann. of Math, 175:1001–1059, 2012.
  * [8] J.E. Hutchinson.  Fractals and self similarity.  Indiana University Mathematics Journal, 30:713–747, 1981.
  * [9] I. Katai and J. Szabo.  Canonical number systems for complex integers.  Acta Sci. Math, 37:255–260, 1975.
  * [10] S. Pedersen and V.T. Shaw.  Dimension of the intersection of certain Cantor sets in the plane.  Opuscula Math, 41:227–244, 2021.
  * [11] P. Shmerkin.  On Furstenberg’s intersection conjecture, self-similar measures, and the $l^{q}$ norms of convolutions.  Ann. of Math, 189:319–391, 2019.
  * [12] P. Walters.  An Introduction to Ergodic Theory.  Springer-Verlag, New York, 1982.
  * [13] M. Wu.  A proof of Furstenberg’s conjecture on the intersections of $\times p$\- and $\times q$-invariant sets.  Ann. of Math, 189:707–751, 2019.

##  Appendix A Derivation of the State Graph ($n\geq 3$)

This appendix is a supplement to the discussion of Figure 1 in Section 2 and
we assume familiarity with that portion of this document. The goal of this
appendix is to demonstrate how Lemma A.1 translates to the state graph in
Figure 1. For convenience, the graph can be found in Figure 2 below and Lemma
A.1 is simply a repetition of Lemma 2.4.

Recall that the claim is that any triple of radix expansions in base-(-n+i)
represent the same complex number if and only if they can be obtained from an
infinite path through the state graph starting from the top node (state). The
diagrams for the states and the labelling system for the edges is the same as
it is in Section 2. Given a radix expansion
$(d_{\ell},d_{\ell-1},\ldots,d_{0};d_{-1},d_{-2},\ldots)$, we use the notation
$d_{k}$ for the $k$th digit. The notation $d(k)$ is the same as it is in
Section 2, but recalling it is unnecessary. It can be treated implicitly
throughout this appendix.

pqrpqrpqrrpqrpqrpqpqrprqqprrqprpqpqrqrp$\scriptsize\begin{matrix}0\\\ 0\\\
0\\\ \end{matrix}$+$\scriptsize\begin{matrix}0\\\ 0\\\ 1\\\
\end{matrix}$+$\scriptsize\begin{matrix}1\\\ 1\\\ 0\\\
\end{matrix}$+$\scriptsize\begin{matrix}1\\\ 0\\\ 2n\\\
\end{matrix}$+$\scriptsize\begin{matrix}2n-1\\\ 2n\\\ 0\\\
\end{matrix}$+$\scriptsize\begin{matrix}0\\\ 1\\\ n^{2}-2n+2\\\
\end{matrix}$+$\scriptsize\begin{matrix}n^{2}-2n+2\\\ n^{2}-2n+1\\\ 0\\\
\end{matrix}$+$\scriptsize\begin{matrix}2n-1\\\ 0\\\ n^{2}\\\
\end{matrix}$$\scriptsize\begin{matrix}n^{2}\\\ 2n-1\\\ 0\\\
\end{matrix}$$\scriptsize\begin{matrix}0\\\ n^{2}\\\ 2n-1\\\
\end{matrix}$$\scriptsize\begin{matrix}n^{2}-2n+1\\\ n^{2}\\\ 0\\\
\end{matrix}$$\scriptsize\begin{matrix}0\\\ n^{2}-2n+1\\\ n^{2}\\\
\end{matrix}$$\scriptsize\begin{matrix}n^{2}\\\ 0\\\ n^{2}-2n+1\\\
\end{matrix}$$\scriptsize\begin{matrix}0\\\ 0\\\ n^{2}-2n+1\\\
\end{matrix}$+$\scriptsize\begin{matrix}0\\\ 0\\\ 2n-1\\\
\end{matrix}$+$\scriptsize\begin{matrix}0\\\ 0\\\ 2n\\\
\end{matrix}$+$\scriptsize\begin{matrix}2n-1\\\ 2n-1\\\ 0\\\
\end{matrix}$+$\scriptsize\begin{matrix}0\\\ 0\\\ n^{2}-2n+2\\\
\end{matrix}$+$\scriptsize\begin{matrix}n^{2}-2n+2\\\ n^{2}-2n+2\\\ 0\\\
\end{matrix}$+$\scriptsize\begin{matrix}2n\\\ 2n\\\ 0\\\
\end{matrix}$+$\scriptsize\begin{matrix}n^{2}-2n+1\\\ n^{2}-2n+1\\\ 0\\\
\end{matrix}$+$\scriptsize\begin{matrix}0\\\ 0\\\ n^{2}\\\
\end{matrix}$$\scriptsize\begin{matrix}n^{2}\\\ n^{2}\\\ 0\\\ \end{matrix}$
Figure 2. The graph governing equivalent radix expansions in base $-n+i$ for
$n\geq 3$.

######  Lemma A.1 (W. J. Gilbert, [5], proposition 1).

Let $n$ be a postive integer. Two radix expansions, $q$ and $r$, represent the
same complex number in base $b=-n+i$ if and only if, for all integers $k$,
either

  * (i)

$q(k)-r(k)\in\\{0,\pm 1,\pm(n+i),\pm(n-1+i)\\}$ when $n\neq 2$, or

  * (ii)

$q(k)-r(k)\in\\{0,\pm 1,\pm(2+i),\pm(1+i),\pm i,\pm(2+2i)\\}$ when $n=2$.

We proceed under the assumption that $n\geq 3$. In [5], Gilbert gives some of
the calculations pertaining to the $n=1$ state graph. The derivation of that
graph does not exhibit all the reasoning featured in the derivation of the
graph governing the cases $n\geq 3$. We discuss the special case of $n=2$ in
Appendix B.

Let $p,q,$ and $r$ be radix expansions in base $b=-n+i$. The $k$th state is
defined to be $S(k):=(p(k)-q(k),q(k)-r(k),r(k)-p(k))$. It is important to
recall that, in this context, the index $k$ ranges the integers and the digit
$p_{k}$ corresponds to the coefficient of $b^{k}$.

Although the sum of the components of $S(k)$ is zero, our notation lists them
all. This is because we wish to explicitly compute the digits of all three
expansions in the $k$th place. We recall the equation ((12) in Section 2)

(39) $S(k)=(p_{k}-q_{k},q_{k}-r_{k},r_{k}-p_{k})+bS(k+1).$

We use the $(k+1)$st state to find the possible values of $S(k)$. Every radix
expansion $d$ has a smallest index $\ell$ at which $d_{k}=0$ for all
$k\geq\ell$. Therefore there exists a $k$ for which $p(k+1)=q(k+1)=r(k+1)=0$
and thus $S(k+1)=(0,0,0)$. This state corresponds with the top node
communicated by the diagram

$\leavevmode\hbox to21.74pt{\vbox
to21.74pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}{}{{{}{}}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{21.33957pt}\pgfsys@lineto{21.33957pt}{21.33957pt}\pgfsys@lineto{21.33957pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{21.33957pt}{21.33957pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.15588pt}{9.48923pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{pqr}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
We compute, using Lemma A.1, the possible values of $S(k)$. Each value will
correspond to a node in the state graph that is a successor of the node
corresponding to $s(k+1)=(0,0,0)$.

Observe that by (39) the $k$th state must satisfy
$S(k)=(p_{k}-q_{k},q_{k}-r_{k},r_{k}-p_{k})$. This forces the components of
$S(k)$ to be integers since each digit is an integer. In accordance with Lemma
A.1, the components must be $0$ or $\pm 1$. This splits into the case that
either all three digits are the same ($S(k)=(0,0,0)$) or at least one digit
differs from the other two.

The case $S(k)=(0,0,0)$ implies the existence of an arrow from the state
$(0,0,0)$ back to itself. The triple of digits $(p_{k},q_{k},r_{k})$ could be
any $(a,a,a)$ where $a\in\\{0,1,\ldots,n^{2}\\}$. This is indicated by the
label on the corresponding edge in the state graph given by

$\begin{matrix}0\\\ 0\\\ 0\end{matrix}+.$

We proceed with the case of differing digits. The digits cannot all be
distinct because this would mean one of the pairs would necessarily have a
difference of magnitude greater than or equal to $2$. Without loss of
generality, let us say that $r$ is the expansion that differs in the $k$th
digit and $p_{k}=q_{k}$. Either $r_{k}$ is one more than $p_{k}$ or one less.
We either have $S(k)=(0,-1,1)$ or $S(k)=(0,1,-1)$. These states correspond to
the diagrams

pqr and rpq

respectively and result in the remaining two edges from the top node present
in the state graph in Figure 2.

The triples $(p_{k},q_{k},r_{k})$ are either of the form $(a,a,a+1)$ or
$(a+1,a+1,a)$ where $a\in\\{0,1,\ldots,n^{2}-1\\}$. This is indicated by the
respective labels

$\begin{matrix}0\\\ 0\\\ 1\end{matrix}+\;\;\text{and}\;\;\begin{matrix}1\\\
1\\\ 0\end{matrix}+$

on the edges in the state graph.

This first step provides the flavour of the calculations that appear in the
full derivation of the graph. We compute a second step which will include the
possibility that all three of the digits $p_{k},q_{k}$, and $r_{k}$ are
distinct. Let us reindex such that $S(k+1)=(0,1,-1)$. Again, we refer to (39)
to direct our calculations. We have

(40) $S(k)=(p_{k}-q_{k},q_{k}-r_{k},r_{k}-p_{k})+(0,-n+i,n-i).$

It is clear that, at least one of the digits must differ from the other two.
Let us investigate the case of exactly one distinct digit. Without loss of
generality we assume $p_{k}=q_{k}$ and $r_{k}\neq p_{k}$. Consider the second
component of $S(k)$: $q_{k}-r_{k}-n+i$.

The digits are integers and thus there is no way of changing the positive
imaginary part. Our only options, according to Lemma A.1, are to either choose
digits $q_{k}$ and $r_{k}$ such that $q_{k}-r_{k}=2n$ or $2n-1$. The choice of
a difference of $2n$ implies that the third component is $-n-i$, which
satisfies Lemma A.1. The resulting state is $S(k)=(0,n+i,-n-i)$. Its
corresonding diagram in Figure 2 is

$\leavevmode\hbox to43.08pt{\vbox
to43.08pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}{}{{{}{}}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{21.33957pt}\pgfsys@lineto{21.33957pt}{21.33957pt}\pgfsys@lineto{21.33957pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{21.33957pt}{21.33957pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{8.71146pt}{8.51701pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{r}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}{}{{{}{}}}{}\pgfsys@moveto{21.33957pt}{21.33957pt}\pgfsys@moveto{21.33957pt}{21.33957pt}\pgfsys@lineto{21.33957pt}{42.67914pt}\pgfsys@lineto{42.67914pt}{42.67914pt}\pgfsys@lineto{42.67914pt}{21.33957pt}\pgfsys@closepath\pgfsys@moveto{42.67914pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{26.45378pt}{30.8288pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{pq}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
The triple of digits $(p_{k},q_{k},r_{k})$ is of the form $(2n+a,2n+a,a)$
where $a\in\\{0,1,\ldots,n^{2}-2n\\}$. This is indicated by the label on the
corresponding edge in the state graph given by

$\begin{matrix}2n\\\ 2n\\\ 0\end{matrix}+.$

If we made the other choice, the resulting state is $S(k)=(0,n-1+i,-n+1-i)$
with the diagram

$\leavevmode\hbox to21.74pt{\vbox
to43.08pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-21.53957pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}{}{{{}{}}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{21.33957pt}\pgfsys@lineto{21.33957pt}{21.33957pt}\pgfsys@lineto{21.33957pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{21.33957pt}{21.33957pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.11421pt}{9.48923pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{pq}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}{}{{{}{}}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{-21.33957pt}\pgfsys@lineto{21.33957pt}{-21.33957pt}\pgfsys@lineto{21.33957pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{21.33957pt}{-21.33957pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{8.71146pt}{-12.82256pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{r}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;\;\text{and with the
label}\;\;\begin{matrix}2n-1\\\ 2n-1\\\ 0\end{matrix}+$

on the incoming edge.

The reasoning above exhausts the case where exactly one of the digits differs.
Now we consider the case where all three are different and, in particular,
$p_{k}\neq q_{k}$. Since the first component of $S(k)$ is precisely
$p_{k}-q_{k}$ (see (40)), it follows from Lemma A.1 that either $p_{k}$ is one
more than $q_{k}$ or one less. The expansions $p$ and $q$ have the same digits
for all places $k+j$ for all $j\geq 1$. We are distinguishing them for the
first time. Without loss of generality we may assume $p_{k}=q_{k}-1$.

In order for the remaining components of $S(k)$ to obey Lemma A.1, we must
have $q_{k}-r_{k}=2n$ and thus $r_{k}-p_{k}=-2n+1$. The resulting state is
$S(k)=(1,n-1+i,-n-i)$ and its corresponding diagram is

$\leavevmode\hbox to43.08pt{\vbox
to43.08pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}{}{{{}{}}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{21.33957pt}\pgfsys@lineto{21.33957pt}{21.33957pt}\pgfsys@lineto{21.33957pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{21.33957pt}{21.33957pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{8.71146pt}{8.51701pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{r}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}{}{{{}{}}}{}\pgfsys@moveto{21.33957pt}{21.33957pt}\pgfsys@moveto{21.33957pt}{21.33957pt}\pgfsys@lineto{21.33957pt}{42.67914pt}\pgfsys@lineto{0.0pt}{42.67914pt}\pgfsys@lineto{0.0pt}{21.33957pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.892pt}{30.8288pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{p}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}{}{{{}{}}}{}\pgfsys@moveto{21.33957pt}{21.33957pt}\pgfsys@moveto{21.33957pt}{21.33957pt}\pgfsys@lineto{21.33957pt}{42.67914pt}\pgfsys@lineto{42.67914pt}{42.67914pt}\pgfsys@lineto{42.67914pt}{21.33957pt}\pgfsys@closepath\pgfsys@moveto{42.67914pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{29.37047pt}{30.8288pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{q}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
The remaining structure of the state graph can be deduced by iterating this
procedure until all the successive states are found. We leave this task to the
interested reader.

##  Appendix B The Other State Graph ($n=2$)

This appendix is a supplement to the discussion of Figure 1 in Section 2. We
assume familiarity with that portion of this document. The goal of this
appendix is to present the state graph governing equivalent radix expansions
in base $-2+i$.

In Lemma 2.4, the difference $p(k)-q(k)$ may take on a larger number of values
when $n=2$. This increases the number of realizable states and thus
complicates the corresponding state graph. The reasoning we employed to derive
the state graph for $n\geq 3$ applies in the case $n=2$ as well. We do not
include the details. We do include the notation required to parse the diagrams
for the new states in the state graph, the primary claim from [5] about the
graph (Theorem B.1), and the graph itself (Figure 3). The new edges particular
to $n=2$ are highlighted in blue and any successor of a blue edge is also a
new state particular to the $n=2$ case.

We make special mention that we only label the edges that correspond to the
first distinction between a pair of expansions. The interested reader can
derive any edge label using the value of the source and successor states of
the edge and (39).

Let $p$ and $q$ be two radix expansions in base $-2+i$. We extend the list of
diagrams from Section 2 that communicate the value of $p(k)-q(k)$. The
additions are as follows:

  1. (v)

$p(k)-q(k)=i$ corresponds to qp.

  2. (vi)

$p(k)-q(k)=2+2i$ corresponds to qp.

We can communicate the value of additional states using these diagrams. For
example, the state $(-1-i,1+i,-2-2i)$ is communicated by the diagram

$\leavevmode\hbox to21.74pt{\vbox
to64.42pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}{}{{{}{}}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{21.33957pt}\pgfsys@lineto{21.33957pt}{21.33957pt}\pgfsys@lineto{21.33957pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{21.33957pt}{21.33957pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{8.71146pt}{8.51701pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{r}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}{}{{{}{}}}{}\pgfsys@moveto{21.33957pt}{21.33957pt}\pgfsys@moveto{21.33957pt}{21.33957pt}\pgfsys@lineto{21.33957pt}{42.67914pt}\pgfsys@lineto{0.0pt}{42.67914pt}\pgfsys@lineto{0.0pt}{21.33957pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{42.67914pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{8.0309pt}{30.8288pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{q}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}{}{{{}{}}}{}\pgfsys@moveto{0.0pt}{42.67914pt}\pgfsys@moveto{0.0pt}{42.67914pt}\pgfsys@lineto{0.0pt}{64.0187pt}\pgfsys@lineto{21.33957pt}{64.0187pt}\pgfsys@lineto{21.33957pt}{42.67914pt}\pgfsys@closepath\pgfsys@moveto{21.33957pt}{64.0187pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.892pt}{52.16837pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{p}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
######  Theorem B.1 (W. J. Gilbert, [5], theorem 8).

Let $p,q$ and $r$ be three radix expansions in base $-2+i$. These expansions
represent the same complex number if and only if they can be obtained from an
infinite path through the state graph in Figure 3 starting at state $(0,0,0)$,
if necessary relabelling $p,q$ and $r$ and in some cases, when $p=q$,
replacing $q$ with another expansion.

pqrpqrpqrrpqrpqrpqpqrprqqprrqprpqpqrqrpqrprqppqrpqrrpqprqpqrqprqrpqrprpqrpq$\scriptsize\begin{matrix}0\\\
0\\\ 1\\\ \end{matrix}$+$\scriptsize\begin{matrix}1\\\ 1\\\ 0\\\
\end{matrix}$+$\scriptsize\begin{matrix}1\\\ 0\\\ 4\\\
\end{matrix}$$\scriptsize\begin{matrix}3\\\ 4\\\ 0\\\
\end{matrix}$$\scriptsize\begin{matrix}0\\\ 1\\\ 2\\\
\end{matrix}$+$\scriptsize\begin{matrix}2\\\ 1\\\ 0\\\
\end{matrix}$+$\scriptsize\begin{matrix}1\\\ 0\\\ 3\\\
\end{matrix}$+$\scriptsize\begin{matrix}2\\\ 3\\\ 0\\\
\end{matrix}$+$\scriptsize\begin{matrix}3\\\ 2\\\ 0\\\
\end{matrix}$+$\scriptsize\begin{matrix}0\\\ 1\\\ 3\\\ \end{matrix}$+ Figure
3. The graph governing equivalent radix expansions in base $-2+i$.