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\begin{document}
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\begin{center}
{\LARGE\bf A Sequential View of Self-Similar Measures;\\
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or, What the Ghosts of Mahler and Cantor\\
\vskip .3cm
Can Teach Us About Dimension}
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\large
Michael Coons and James Evans\\
School of Mathematical and Physical Sciences\\
The University of Newcastle\\
Callaghan, NSW 2308\\
Australia\\
\href{mailto:Michael.Coons@newcastle.edu.au}{\tt Michael.Coons@newcastle.edu.au}\\
\href{mailto:James.Evans10@uon.edu.au}{\tt James.Evans10@uon.edu.au}
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\begin{abstract} We show that a missing $q$-ary digit set
$F\subseteq[0,1]$ has a corresponding naturally associated countable
binary $q$-automatic sequence $f$. Using this correspondence, we show
that the Hausdorff dimension of $F$ is equal to the base-$q$ logarithm
of the Mahler eigenvalue of $f$. In addition, we demonstrate that the
standard mass distribution $\nu_F$ supported on $F$ is equal to the
ghost measure $\mu_f$ of $f$.
\end{abstract}
\section{Introduction}
This story starts with an uncountable set and a countable sequence, their respective dimension and asymptotical behavior, and describes the setting in which these objects and properties---and their generalizations---coalesce.
The uncountable set is the ubiquitous standard middle-thirds Cantor set, $C$, the self-affine set that is the unique attractor of the iterated function system $\mathcal{S}_C:=\{S_1,S_2\}$, where the functions $S_1,S_2:[0,1]\to[0,1]$ are the affine contractions $$S_1(x)=\frac{x}{3}\qquad\mbox{and}\qquad S_2(x)=\frac{2+x}{3}.$$ The standard construction of $C$ is via the production of ``level sets'' $E_k$, for $k\geq 0$, where $E_k:=S_C^k([0,1])$, $S_C(E):=S_1(E)\cup S_2(E)$ for any set $E$, and $S_C^k$ denotes $k$-fold composition of $S_C$ with itself; see Figure \ref{fig:cant}. Each level set $E_k$ has Lebesgue measure $\lambda(E_k)=(2/3)^k$, so that $C$ has $\lambda$-measure zero. Nonetheless, $C$ has some size; it has Hausdorff dimension $\log_3(2)$, where $\log_3(\cdot)$ denotes the ternary logarithm. Further, one can construct a natural mass distribution $\nu_C$, called the Cantor measure, supported on $C$ by repeated subdivision, at each level spreading the mass equally among the subintervals of $E_k$ and taking $k$ to infinity. In this way, the $\lambda$-singular continuous measure $\nu_C$ is the (weak) limit of $\lambda$-absolutely continuous measures. We thus arrive to the paradigmatic example of a fractal set $C$ and a related mass distribution $\nu_C$. Much of the theories of fractal geometry and discrete dynamical systems have been developed pursuing generalizations of this example; see Falconer's definitive monograph {\em Fractal Geometry} \cite{Falconer} for details and discussions.
\begin{figure}[hb]
\begin{center}
\begin{tikzpicture}
\foreach \order in {0,...,5,6}
\draw[yshift=-\order*17pt] l-system[l-system={cantor set, axiom=F, order=\order, step=300pt/(3^\order)}];
\put(0,2){{\tiny $0$}}
\put(98,4){{\tiny $\frac{1}{3}$}}
\put(198,4){{\tiny $\frac{2}{3}$}}
\put(298,2){{\tiny $1$}}
\put(305,-3){{$E_0$}}
\put(305,-20){{$E_1$}}
\put(305,-37){{$E_2$}}
\put(305,-54){{$E_3$}}
\put(305,-71){{$E_4$}}
\put(307,-90){{$\vdots$}}
\put(305,-105){{$C$}}
\draw[white,line width=10pt] (0,-3) -- (10.55,-3);
\end{tikzpicture}
\end{center}
\label{fig:cant}
\caption{Iterated construction of the Cantor set $C$, where $E_k=S_C^k([0,1])$.}
\end{figure}
Transitioning from the uncountable to the countable, we consider the $3$-automatic binary sequence that is the infinite fixed point $\varrho_c^\infty(1)$ of the substitution $\varrho_{c}$ defined by
\begin{equation*}
\varrho_{c}: \, \begin{cases} 1 \mapsto 101 \\
0\mapsto 000\, . \end{cases}
\end{equation*} We define the {\em Cantor sequence} $\{c(n)\}_{n\geq 0}$ as the $n$th entry of $\varrho_c^\infty(1)$. The construction of $\varrho_c^\infty(1)$ is reminiscent of the construction of the Cantor set $C$, the iterate $\varrho_c^k(1)$ playing the part of the level set $E_k$; see Figure \ref{fig:seq}. It is easy to see that $c(n)=1$ precisely when the ternary expansion of the integer $n$ contains no ones, so that analogy to the Cantor set is clear. See Allouche and Shallit \cite{ASbook} for details on automatic sequences.
\begin{figure}[h]
\begin{align*}
\varrho_{\rm c}^0(1)&= 1\\
\varrho_{\rm c}^1(1)&= 101\\
\varrho_{\rm c}^2(1)&= 101000101\\
\varrho_{\rm c}^3(1)&= 101000101000000000101000101\\
&\ \, \vdots\\
\varrho_{\rm c}^\infty(1)&=
101000101000000000101000101000000000000000000000000000101000\cdots
\end{align*}
\caption{Iterated construction of the Cantor sequence $c$}
\label{fig:seq}
\end{figure}
Using the definition, since $c$ is $3$-automatic, its generating function is a Mahler function, and one obtains immediately that the generating function of the Cantor sequence can be written as an infinite product, $$M_c(z):=\sum_{n\geq 0}c(n)z^n=\prod_{j\geq 0}\left(1+z^{2\cdot 3^j}\right).$$ A result of Bell and Coons \cite{BC2017} provides, as $z\to 1^-$, the asymptotics $$M_c(z)\asymp\frac{1}{(1-z)^{\log_3(2)}}(1+o(1)).$$ Here, the number $2$ in the quantity $\log_3(2)$ is the Mahler eigenvalue $\lambda_c$ of $M_c(z)$ (details and definitions are provided in the next section) obtained from the Mahler-type functional equation satisfied by $M_c(z)$, $$M_c(z)-(1+z^2)M_c(z^3)=0.$$ Note that the divergent behavior of $M_c(z)$ is governed by the Hausdorff dimension of $C$; that is, $\dim_H C=\log_3(2)$. This relationship, generalized, is our first result.
\begin{theorem}\label{main1} Let $q\geq 2$ be an integer, $A\subseteq\{0,\ldots,q-1\}$ containing $0$ and $F$ be the subset of $[0,1]$ consisting of numbers that can be represented in base-$q$ using only $q$-ary digits from $A$. Then, with $F$ there is a naturally associated $q$-automatic binary sequence $f$ such that \begin{equation}\label{eqthm1}\dim_{\rm H}F=\log_q(|A|)=\log_q(\lambda_{f}),\end{equation} where $\lambda_{f}$ is the Mahler eigenvalue of the generating function $M_f(z)$ of $f$ and $\log_q(\cdot)$ denotes the base-$q$ logarithm.
\end{theorem}
\begin{remark} We note that the new contribution of Theorem \ref{main1} concerns the connection to the Mahler eigenvalue. In particular, the second equality of \eqref{eqthm1} is new---the first follows directly from Falconer \cite[Thm.~9.3]{Falconer}.
\end{remark}
Returning to our motivating example, analogous to the Cantor measure $\nu_C$, we may construct a mass distribution $\mu_c$ associated with the sequence $c$, called the ghost\footnote{The term {\em ghost measure} was introduced by the second author \cite{EvansPRE}, inspired by the phrase ``the ghost of departed quantities'' as appearing in Berkeley's critique of calculus. Evans \cite{EvansPRE} writes, ``The terms of the sequence are (usually) much smaller than the sum of the terms, so the individual pure points of the $\mu_n$ are driven to zero by the averaging as $n$ tends to infinity. The measure $\mu$ is the ghost of the departed pure points of the $\mu_n$.''} measure. This construction was introduced by Baake and Coons \cite{BCaa}. Here, for each $n$, we take the sequence $c$ up to $3^n-1$ (a kind of `fundamental region') and reinterpret its (renormalized) values as the weights of a pure point probability measure $\mu_{c,n}$ on the torus $\mathbb{T}=[0,1)$ with support ${\rm supp}(\mu_{c,n})=\left\{\frac{j}{3^n}:0\leq j<3^n,\, c(j)=1\right\}$. That is, $$\mu_{c,n}:=2^{-n}\sum_{j=0}^{3^n-1}c(j)\, \delta_{j/3^n},$$ where $\delta_x$ denotes the unit Dirac measure at $x$. The ghost measure $\mu_c$ is the (weak) limit of the sequence $(\mu_{c,n})_{n\geq 0}$ as $n\to\infty$. With the obvious generalization in notation, we state our second result.
\begin{theorem}\label{main2} With the notation above and assumptions of Theorem \ref{main1}, the standard mass distribution $\nu_F$ supported on $F$ is equal to the ghost measure $\mu_f$.
\end{theorem}
\noindent In particular, the Cantor measure $\nu_C$ is equal to the ghost measure $\mu_c$. In this way, Theorem~\ref{main2} provides an alternative construction of standard mass distributions supported on fractals, and does so in a way that takes the countable to the uncountable.
\begin{remark} Self-similar sets---attractors of affine contractions---have been used by many authors to model physical phenomena; see Mandelbrot \cite{M1977} for a detailed history and references. The study of self-similar sets was put into a rigorous general framework by Hutchinson \cite{H1981} and the study of self-similar measures and their Fourier transforms was studied in detail by Strichartz \cite{S1990,S1993}; see also Makarov \cite{M1993,M1994}. See Baake and Moody \cite{BM2000} for the generalization to compact families of contractions.
\end{remark}
\section{Hausdorff dimension and the Mahler eigenvalue}
In this section, we prove Theorem \ref{main1}.
To this end, let $q\geq 2$ be an integer, $A\subseteq\{0,\ldots,q-1\}$ containing $0$ and $F$ be the subset of $[0,1]$ consisting of numbers that can be represented in base-$q$ using only $q$-ary digits form $A$. We enumerate the set $A$ as $$0=a_1