# Purity results for some arithmetically defined measures

Peter J. Grabner* Institut für Analysis und Zahlentheorie, Technische
Universität Graz, Kopernikusgasse 24/II, 8010 Graz, Austria
<EMAIL_ADDRESS>Dedicated to Jörg Thuswaldner on the occasion of his
50th birthday.

###### Abstract.

We study measures that are obtained as push-forwards of measures of maximal
entropy on sofic shifts under digital maps
$(x_{k})_{k\in\mathbb{N}}\mapsto\sum_{k\in\mathbb{N}}x_{k}\beta^{-k}$, where
$\beta>1$ is a Pisot number. We characterise the continuity of such measures
in terms of the underlying automaton and show a purity result.

###### 2010 Mathematics Subject Classification:

11A63, 68Q45, 11K55, 37A45

* The author is supported by the Austrian Science Fund FWF project F5503 (part of the Special Research Program (SFB) “Quasi-Monte Carlo Methods: Theory and Applications”)

##  1\. Introduction

Digital representations of real numbers by infinite series

(1) $\sum_{k=1}^{\infty}\frac{x_{k}}{\beta^{k}}$

with $x_{k}\in\mathcal{A}$, a finite alphabet, and $\beta>1$ have attracted
attention from different points of view. The underlying dynamical system given
by the map $T_{\beta}:x\mapsto\beta x\mod 1$ has been studied extensively
since the seminal papers [32, 26]. For an overview of the development we refer
to [8, 16, 15, 3]. The original study was carried out for the “canonical”
digit set $\mathcal{A}=\\{0,1,\ldots,\lceil\beta\rceil-1\\}$, but many
variations have been studied. It turned out in [26] that Pisot numbers $\beta$
play a very important rôle in that context, as for these $\beta$ the
transformation $T_{\beta}$ has especially nice properties. In this case the
set of representations of all real numbers in $[0,1]$ obtained by iteration of
$T_{\beta}$ is a sofic shift (see [26]); the definition of a sofic shift will
be given in Section 2. A Pisot number is an algebraic integer all of whose
Galois conjugates have modulus $<1$ (see [4]). Notice that integers $\geq 2$
are also considered as Pisot numbers. Pisot numbers have the nice property
that their powers are “almost integers”, meaning that $(\beta^{n}\mod
1)_{n\in\mathbb{N}}$ tends to $0$.

In the present paper we will change the point of view starting with a one-
sided sofic shift space $\mathcal{K}^{+}\subset\mathcal{A}^{\mathbb{N}}$,
where $\mathcal{A}\subset\mathbb{Z}$ is the underlying set of digits. Then we
consider a map $\phi^{+}:\mathcal{K}^{+}\to\mathbb{R}$ mapping
$(x_{k})_{k\in\mathbb{N}}$ to the series (1) for $\beta$ a Pisot number. Of
course, in general nothing can be said about injectivity of this map, or even
the structure of the image. Even for the full shift $\mathcal{A}^{\mathbb{N}}$
the structure of the image can be intricate (see [45]). If $\mathcal{K}^{+}$
is equipped with a shift invariant measure, then this measure is pushed
forward to $\mathbb{R}$ by the map $\phi^{+}$. The properties of measures
obtained in this way are our object of study. For the measure on
$\mathcal{K}^{+}$ we will take the measure of maximal entropy, or Parry
measure, see [28, 27]. This will be discussed in Section 3.

Measures of this kind occur in different contexts. Possibly, the earliest
occurrence was in two papers by Erdős [13, 12], where he proved the
singularity of this measure for $\mathcal{K}$ being the full shift $\\{\pm
1\\}^{\mathbb{N}}$. This is the most studied case and goes under the name
“Bernoulli convolution”. We will discuss that further in Section 4.

In the context of studying redundant expansions of integers in the context of
fast multiplication algorithms used in cryptography, the precise study of the
number of representations of an integer $n$ in the form

$n=\sum_{k=0}^{K}x_{k}2^{k}\quad\text{with }x_{k}\in\\{0,\pm 1\\}$

and minimising

$\sum_{k=0}^{K}|x_{k}|$

led to a singular measure on $[-1,1]$ (see [17]). Further results in this
direction were obtained in [19, 18]. A more general point of view replacing
the powers of $2$ by the solution of a linear recurrence has been taken in
[21]. Furthermore, such measures occur as spectral measures of dynamical
systems related to numeration systems [20], in the study of diffraction
patters of tilings (see [2, 1]), and as spectral measures of substitution
dynamical systems (see [30]). In Section 5 we will present two main results,
namely the fact that the measure is pure (meaning that the Lebesgue
decomposition only has one term), and a characterisation of continuity of the
measure in terms of properties of the underlying automaton.

Erdős’ proof of the singularity of the measure in [13, 12] uses the fact that
the Fourier transform of these measures does not tend to $0$ at $\infty$, and
this method was used in many other cases. This motivates the study of the
Fourier transform of the measures under consideration. The transform can be
expressed in terms of infinite matrix products, which allow the computation of
limits

(2) $\lim_{k\to\infty}\widehat{\nu}(z\beta^{k})$

for $z\in\mathbb{Z}[\beta]$. In Section 6 we find an interpretation of these
limits as Fourier coefficients of a measure on the torus given by a two sided
version of the map $\phi^{+}$. A similar map has been introduced and studied
in [39, 38]. This will be used to show that the vanishing of the limits (2)
for all $z\in\mathbb{Z}[\beta]$, $z\neq 0$ is equivalent to absolute
continuity for these measures.

Very recently, similar results for self similar measures have been obtained in
[5]. There for a set of linear maps $\phi_{k}:x\mapsto r_{k}x+b_{k}$ with
$0<r_{k}<1$ ($k=1,\ldots,N$) the distribution measure $\nu$ of the map

$(\omega_{1},\omega_{2},\ldots)\mapsto\lim_{n\to\infty}\phi_{\omega_{n}}\circ\phi_{\omega_{n-1}}\circ\cdots\circ\phi_{\omega_{1}}(x_{0})$

is studied, where $\\{1,\ldots,N\\}^{\mathbb{N}}$ is equipped with the
infinite product of the measures $\mathbb{P}(\\{k\\})=p_{k}$ ($k=1,\ldots,N$)
for any choice of the vector $(p_{1},\ldots,p_{N})$ with $p_{k}>0$ and
$\sum_{k}p_{k}=1$. It is shown (see [5, Theorem 2.3]) that
$\widehat{\nu}(t)\not\to 0$ for $|t|\to\infty$, if and only if the contraction
factors $r_{k}$ are all negative integer powers of the same Pisot number. It
is also shown (see [5, Theorem 2.4]) that for these measures
$\lim_{|t|\to\infty}\widehat{\nu}(t)=0$ (called the _Rajchman property_) is
equivalent to absolute continuity. This is a phenomenon very similar to
Theorem 5 in the context of this paper.

In a final Section 7 we exhibit several simple examples as applications of our
results.

##  2\. Regular languages and sofic shifts

Let $\mathcal{A}\subset\mathbb{Z}$ be a finite alphabet, and $G=(V,E)$ be a
finite directed graph (see [9]) equipped with a _labelling_
$\ell:E\to\mathcal{A}$. Then the pair $(G,\ell)$ is called a _labelled graph_.
A finite automaton is a quadruple $\mathsf{A}=(G,\ell,I,T)$, where $I$
(initial states) and $T$ (terminal states) are subsets of the set of vertices
(also called _states_ in this context). A _path_ of length $n$ in the graph
$G$ is a sequence of edges $p=e_{1}e_{2}\ldots e_{n}$, such that for every
$j=1,\ldots,n-1$ the edges $e_{j}=(\mathrm{i}(e_{j}),\mathrm{t}(e_{j})))$
satisfy $\mathrm{t}(e_{j})=\mathrm{i}(e_{j+1})$; the terminal vertex
$\mathrm{t}(e_{j})$ of every edge coincides with the initial vertex
$\mathrm{i}(e_{j+1})$ of the consecutive edge. We say that $p$ connects
$\mathrm{i}(e_{1})$ and $\mathrm{t}(e_{n})$. The language
$\mathcal{L}=\mathcal{L}(\mathsf{A})$ recognised by the automaton $\mathsf{A}$
is given by

(3)
$\begin{split}\mathcal{L}_{n}&=\left\\{\ell(e_{1})\ell(e_{2})\ldots\ell(e_{n})\,\middle|\,e_{1}e_{2}\ldots
e_{n}\text{ a path in }G,\mathrm{i}(e_{1})\in I,\mathrm{t}(e_{n})\in
T\right\\}\\\
\mathcal{L}&=\\{\epsilon\\}\cup\bigcup_{n=1}^{\infty}\mathcal{L}_{n},\end{split}$

where $\epsilon$ denotes the _empty word_ , which by definition has length
$0$; $\mathcal{L}_{n}$ is the set of words of length $n$. A subset of
$\mathcal{A}^{*}$ is called a _regular language_ , if it is recognised by a
finite automaton $\mathsf{A}$. A language $\mathcal{L}$ is called
_irreducible_ , if for any $w_{1},w_{2}\in\mathcal{L}$ there exists a
$w\in\mathcal{A}^{*}$ such that $w_{1}ww_{2}\in\mathcal{L}$. This is
equivalent to the underlying graph being _strongly connected_ ; for any two
vertices $v_{1},v_{2}\in V$ there is a path connecting $v_{1}$ with $v_{2}$.
The language is called _primitive_ , if there is an $N\in\mathbb{N}$ such that
for any $w_{1},w_{2}\in\mathcal{L}$ and any $n\geq N$ there exists a word
$w\in\mathcal{A}^{*}$ of length $n$ such that $w_{1}ww_{2}\in\mathcal{L}$.

From now on we assume that all languages are primitive. For a comprehensive
introduction to the theory of formal languages we refer to [10, 22, 34].

For a given automaton $\mathsf{A}$ we study the sets of one- and two-sided
infinite words recognised by $\mathsf{A}$

(4)
$\begin{split}\mathcal{K}_{I}^{+}&=\left\\{(\ell(e_{k}))_{k\in\mathbb{N}}\,\middle|\,\forall
n\in\mathbb{N}:e_{1}e_{2}\ldots e_{n}\text{ is a path in
}G,\mathrm{i}(e_{1})\in I\right\\}\\\
\mathcal{K}^{+}&=\left\\{(\ell(e_{k}))_{k\in\mathbb{N}}\,\middle|\,\forall
n\in\mathbb{N}:e_{1}e_{2}\ldots e_{n}\text{ is a path in }G\right\\}\\\
\mathcal{K}&=\left\\{(\ell(e_{k}))_{k\in\mathbb{Z}}\,\middle|\,\forall
m<n:e_{m}e_{m+1}\ldots e_{n}\text{ is a path in }G\right\\}.\end{split}$

The set $\mathcal{K}$ is called the (two-sided) _sofic_ shift associated to
$\mathsf{A}$ (see [24]), $\mathcal{K}^{+}$ is the one-sided sofic shift, both
spaces are closed under the according shift transformation

$\displaystyle\sigma^{+}$
$\displaystyle:\mathcal{K}^{+}\to\mathcal{K}^{+}:(x_{k})_{k\in\mathbb{N}}\mapsto(x_{k+1})_{k\in\mathbb{N}}$
$\displaystyle\sigma$
$\displaystyle:\mathcal{K}\to\mathcal{K}:(x_{k})_{k\in\mathbb{Z}}\mapsto(x_{k+1})_{k\in\mathbb{Z}}.$

The space $\mathcal{K}_{I}^{+}$, which can be seen as an extension of
$\mathcal{L}$ to infinite words, is in general not closed under $\sigma^{+}$,
but the relation

$\mathcal{K}^{+}=\bigcup_{n=0}^{N}(\sigma^{+})^{n}\mathcal{K}_{I}^{+}$

holds for some $N\in\mathbb{N}$ as a consequence of the strong connectedness
of the graph underlying $\mathsf{A}$. Notice that $\sigma$ is bijective,
whereas $\sigma^{+}$ is not.

##  3\. Measures on shift spaces

We equip the spaces $\mathcal{K}_{I}^{+}$ and $\mathcal{K}^{+}$ with a
“canonical” measure that we will define now. We define the _cylinder set_

$[\varepsilon_{1},\varepsilon_{1},\ldots,\varepsilon_{k}]=\left\\{(x_{n})_{n\in\mathbb{N}}\in\mathcal{K}_{I}^{+}\,\middle|\,x_{1}=\varepsilon_{1},\ldots,x_{k}=\varepsilon_{k}\right\\}$

for $\varepsilon_{i}\in\mathcal{A}$ for $i=1,\ldots,k$. The cylinder sets
generate a topology on $\mathcal{K}_{I}^{+}$ and also the $\sigma$-algebra of
Borel sets for this topology. We define

(5)
$\mu_{I}^{+}([\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{k}])=\lim_{n\to\infty}\frac{\\#([\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{k}]\cap\mathcal{L}_{n})}{\\#\mathcal{L}_{n}};$

the existence of the limit will become obvious from the following discussion.

For $a\in\mathcal{A}$ define the _$a$ -transition matrix_ by

(6) $(M_{a})_{ij}=\begin{cases}1&\text{if }(i,j)\in E\text{ and
}\ell((i,j))=a\\\ 0&\text{otherwise}\end{cases}$

and

(7) $M=\sum_{a\in\mathcal{A}}M_{a}.$

Furthermore, set $\mathbf{v}_{I}$ and $\mathbf{v}_{T}$ the indicator vectors
of the sets $I$ and $T$, respectively. Then for $n\geq k$ we have

$\\#([\varepsilon_{1},\varepsilon_{1},\ldots,\varepsilon_{k}]\cap\mathcal{L}_{n})=\mathbf{v}_{I}^{\mathsf{T}}M_{\varepsilon_{1}}\cdots
M_{\varepsilon_{k}}M^{n-k}\mathbf{v}_{T}.$

By the assumption that the language $\mathcal{L}$ is primitive which is
equivalent to the fact that $M$ is primitive, the Perron-Frobenius theorem
(see[35]) implies that there is a dominating eigenvalue $\lambda>0$ such that

(8)
$M^{n}=\lambda^{n}\mathbf{v}_{R}\mathbf{v}_{L}^{\mathsf{T}}+o(\lambda^{n}),$

where $\mathbf{v}_{L}^{\mathsf{T}}$ is a left eigenvector of $M$ for the
eigenvalue $\lambda$, and $\mathbf{v}_{R}$ is a right eigenvector with

$\mathbf{v}_{L}^{\mathsf{T}}\mathbf{v}_{R}=1.$

With this we can write the quantity under the limit in (5) as

$\frac{\mathbf{v}_{I}^{\mathsf{T}}M_{\varepsilon_{1}}\cdots
M_{\varepsilon_{k}}\left(\lambda^{n-k}\mathbf{v}_{R}\mathbf{v}_{L}^{\mathsf{T}}+o(\lambda^{n})\right)\mathbf{v}_{T}}{\mathbf{v}_{I}^{\mathsf{T}}\left(\lambda^{n}\mathbf{v}_{R}\mathbf{v}_{L}^{\mathsf{T}}+o(\lambda^{n})\right)\mathbf{v}_{T}},$

which shows that the limit exists and equals

(9)
$\mu_{I}^{+}([\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{k}])=\lambda^{-k}\frac{\mathbf{v}_{I}^{\mathsf{T}}M_{\varepsilon_{1}}\cdots
M_{\varepsilon_{k}}\mathbf{v}_{R}}{\mathbf{v}_{I}^{\mathsf{T}}\mathbf{v}_{R}}.$

On $\mathcal{K}^{+}$ we define the measure $\mu^{+}$ by

(10)
$\mu^{+}([\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{k}])=\lambda^{-k}\mathbf{v}_{L}^{\mathsf{T}}M_{\varepsilon_{1}}\cdots
M_{\varepsilon_{k}}\mathbf{v}_{R}.$

This measure is shift invariant by the observation

$\mu^{+}((\sigma^{+})^{-1}[\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{k}])=\sum_{a\in\mathcal{A}}\mu^{+}([a,\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{k}])\\\
=\sum_{a\in\mathcal{A}}\lambda^{-k-1}\mathbf{v}_{L}^{\mathsf{T}}M_{a}M_{\varepsilon_{1}}\cdots
M_{\varepsilon_{k}}\mathbf{v}_{R}=\lambda^{-k-1}\mathbf{v}_{L}^{\mathsf{T}}MM_{\varepsilon_{1}}\cdots
M_{\varepsilon_{k}}\mathbf{v}_{R}\\\
=\lambda^{-k}\mathbf{v}_{L}^{\mathsf{T}}M_{\varepsilon_{1}}\cdots
M_{\varepsilon_{k}}\mathbf{v}_{R}=\mu^{+}([\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{k}]).$

A similar computation shows that the measure is well defined by Kolmogorov’s
consistency theorem.

Assuming that $\mathcal{L}$ is primitive, the dynamical system
$(\mathcal{K}^{+},\mu^{+},\sigma^{+})$ is strongly mixing and thus ergodic
(see [44, 7]):

$\lim_{n\to\infty}\mu^{+}\left([\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{k}]\cap(\sigma^{+})^{-n}[\delta_{1},\ldots,\delta_{s}]\right)\\\
=\lim_{n\to\infty}\lambda^{-n-s}\mathbf{v}_{L}^{\mathsf{T}}M_{\varepsilon_{1}}\cdots
M_{\varepsilon_{k}}M^{n-k}M_{\delta_{1}}\cdots M_{\delta_{s}}\mathbf{v}_{R}\\\
=\lambda^{-k}\mathbf{v}_{L}^{\mathsf{T}}M_{\varepsilon_{1}}\cdots
M_{\varepsilon_{k}}\mathbf{v}_{R}\lambda^{-s}\mathbf{v}_{L}^{\mathsf{T}}M_{\delta_{1}}\cdots
M_{\delta_{s}}\mathbf{v}_{R}\\\
=\mu^{+}([\varepsilon_{1},\varepsilon_{1},\ldots,\varepsilon_{k}])\mu^{+}([\delta_{1},\ldots,\delta_{s}]).$

The measures $\mu_{I}^{+}$ and $\mu^{+}|_{\mathcal{K}_{I}^{+}}$ (restriction
to $\mathcal{K}_{I}^{+}$) are equivalent. The measure $\mu^{+}$ is the unique
measure of maximal entropy on $\mathcal{K}^{+}$, also called the Parry measure
[28, 27].

Similarly, we define a measure on $\mathcal{K}$ by

(11)
$\mu([\varepsilon_{1},\ldots,\varepsilon_{k}]_{m})=\lambda^{-k}\mathbf{v}_{L}^{\mathsf{T}}M_{\varepsilon_{1}}\cdots
M_{\varepsilon_{k}}\mathbf{v}_{R},$

where

$[\varepsilon_{1},\ldots,\varepsilon_{k}]_{m}=\left\\{(x_{n})_{n\in\mathbb{Z}}\in\mathcal{K}\,\middle|\,x_{m+1}=\varepsilon_{1},\ldots,x_{m+k}=\varepsilon_{k}\right\\}$

for $m\in\mathbb{Z}$. The measure $\mu$ is $\sigma$-invariant by definition.

##  4\. Bernoulli convolutions

In [12] Erdős studied the distribution measure of the random series

(12) $\sum_{k=1}^{\infty}\frac{X_{k}}{\beta^{k}},$

where $(X_{k})_{k\in\mathbb{N}}$ is a sequence of i.i.d. random variables
taking the values $\pm 1$ with equal probability $\frac{1}{2}$, and
$\beta=\frac{1+\sqrt{5}}{2}$. He showed that the distribution is purely
singular continuous. Later [13] he extended this result for $\beta$ an
irrational Pisot number. The Pisot property plays an important rôle in the
proof of singularity, as it allows to show that the Fourier transform of the
measure does not tend to $0$ along the sequence
$(\beta^{k})_{k\in\mathbb{N}}$. This argument will be elaborated later.

In the meantime the set of $\beta>1$, for which the measure constructed as
above is singular continuous has been studied further. Solomyak [40] could
prove that for almost all $\beta\in(1,2)$ the measure is absolutely
continuous. This result was refined by Shmerkin [36], who proved that the
exceptional set has Hausdorff dimension $0$. It is still open, whether Pisot
numbers are the only exceptions. For a survey on the development until the
year 2000 we refer to [29]. For more recent developments and results in this
direction we refer to [33, 37].

A newer development in the study of Bernoulli convolutions was initiated with
the proof that ergodic invariant measures on the full shift
$\mathcal{A}^{\mathbb{N}}$ are projected to exact dimensional measures by
iterated function systems (see [14]). More precisely, a measure $\nu$ is
called exact dimensional with dimension $\alpha$, if

$\lim_{r\to 0}\frac{\log\nu(x-r,x+r)}{\log r}=\alpha$

for $\nu$-almost all $x$. Of course, absolutely continuous measures on
$\mathbb{R}$ have dimension $1$, whereas the opposite implication is not true.
This allowed for the proof that the dimension of the distribution measure of
(12) has dimension $1$ for all transcendental $\beta\in(1,2)$ (see [42]). For
further results in that direction we refer to [6, 41, 43].

##  5\. Generalised Erdős measures

From now on we assume that the alphabet $\mathcal{A}$ is a subset of
$\mathbb{Z}$. For an automaton $\mathsf{A}$ and a Pisot number $\beta$ of
degree $r\geq 1$ we introduce the maps

(13)
$\begin{split}\phi_{\mathcal{L}}^{+}&:\mathcal{L}\to\mathbb{R}:(x_{k})_{k=1}^{n}\mapsto\sum_{k=1}^{n}\frac{x_{k}}{\beta^{k}}\\\
\phi_{I}^{+}&:\mathcal{K}_{I}^{+}\to\mathbb{R}:(x_{k})_{k\in\mathbb{N}}\mapsto\sum_{k=1}^{\infty}\frac{x_{k}}{\beta^{k}}\\\
\phi^{+}&:\mathcal{K}^{+}\to\mathbb{R}:(x_{k})_{k\in\mathbb{N}}\mapsto\sum_{k=1}^{\infty}\frac{x_{k}}{\beta^{k}}.\end{split}$

The measures

(14) $\begin{split}\nu_{I}&=\left(\phi_{I}^{+}\right)_{*}(\mu_{I}^{+})\\\
\nu&=\left(\phi^{+}\right)_{*}(\mu^{+})\end{split}$

are analogues of the Erdős measures studied in [13, 12]. Here and throughout
this paper we denote by $f_{*}(\mu)$ the push-forward measure on $Y$ given by
a map $f:X\to Y$ and a measure $\mu$ on $X$; $f_{*}(\mu)(A)=\mu(f^{-1}(A))$.
The properties of the measures $\nu$ and $\nu_{I}$ will be the subject of the
remaining part of this paper. By definition, $\nu_{I}$ is absolutely
continuous with respect to $\nu$. Furthermore, by the definition of $\mu_{I}$,
$\nu_{I}$ is given by

$\nu_{I}=\lim_{n\to\infty}\frac{1}{\\#\mathcal{L}_{n}}\sum_{w\in\mathcal{L}_{n}}\delta_{\phi_{\mathcal{L}}^{+}(w)},$

where $\delta_{x}$ denotes a unit point mass in $x$.

######  Theorem 1.

The measures $\nu_{I}$ and $\nu$ are pure in the sense that they are either
absolutely continuous with respect to Lebesgue measure, purely singular
continuous, or purely atomic. The last case can only occur, if the image
$\phi^{+}(\mathcal{K}^{+})$ is finite. The number of atoms is bounded by the
number of vertices in $\mathsf{A}$.

###### Proof.

The Jessen-Wintner theorem [23, Theorem 35] (for a more modern formulation see
also [11, Lemma 1.22 (ii)]) is concerned with the distribution measure of a
random series

$\sum_{n=1}^{\infty}X_{n},$

where $(X_{n})_{n\in\mathbb{N}}$ is a sequence of independent discrete random
variables and the series

$\sum_{n=1}^{\infty}\mathbb{E}(X_{n})\text{ and
}\sum_{n=1}^{\infty}\mathbb{V}(X_{n})$

converge. It states that this measure is either absolutely continuous with
respect to Lebesgue measure, purely singular continuous, or purely atomic.

For our purposes we need a more general version, which allows allows some
dependence between the random variables $X_{n}$.

######  Lemma 1.

Let $\mathcal{K}^{+}\subset\mathcal{A}^{\mathbb{N}}$ be a shift invariant
subset equipped with a shift invariant measure $\mu$ such that the shift is
ergodic with respect to $\mu$. Let

$X=\sum_{n=1}^{\infty}X_{n}$

be a series of random variables $X_{n}$, where $X_{n}$ only depends on the
$n$-th coordinate of the argument. Also assume that the series converges
$\mu$-almost everywhere. Then the distribution of $X$ is either purely
discrete, or purely singular continuous, or absolutely continuous with respect
to Lebesgue measure.

###### Proof of Lemma 1.

The proof of this lemma is just the observation that the proof of the Jessen-
Wintner theorem only uses the fact that the measure on the product space
satisfies a $0$-$1$-law. The proof follows the lines of proof given for [11,
Lemma 1.22(ii)]. Let $\nu=X_{*}(\mu)$ be the distribution measure of $X$. Then
$\nu$ has a Lebesgue decomposition

$\nu=a_{1}\nu_{1}+a_{2}\nu_{2}+a_{3}\nu_{3},$

where $a_{1}+a_{2}+a_{3}=1$ and $\nu_{i}$ ($i=1,2,3$) are probability
measures, where $\nu_{1}$ is purely atomic, $\nu_{2}$ is absolutely
continuous, and $\nu_{3}$ is singular continuous. Let $D$ be the support of
$\nu_{1}$. Then $X^{-1}(D)$ is a shift invariant subset of $\mathcal{K}^{+}$,
which has measure $0$ or $1$ by ergodicity. Thus $a_{1}$ is either $0$ or $1$.
Similarly, let $S$ be the set of Lebesgue measure $0$ that supports $\mu_{3}$.
Then again $X^{-1}(S)$ is shift invariant, and again has measure $0$ or $1$.
Thus the decomposition has exactly one term. ∎

The statement for $\nu_{I}$ follows from the absolute continuity of $\nu_{I}$
with respect to $\nu$. The assertion about the number of atoms will be proved
in the proof of Theorem 2. ∎

######  Lemma 2.

Let $P=\\{x\in\mathbb{R}\mid\nu(\\{x\\}))>0$. Then $P$ is a finite subset of
$\mathbb{Q}(\beta)$.

###### Proof.

For $x\in P$ set $A_{x}=(\phi^{+})^{-1}(\\{x\\})$. Then
$\phi^{+}(A_{x})=\\{x\\}$ and

$\phi^{+}(\sigma^{-m}(A_{x}))\subseteq\left\\{\beta^{-m}\left(x+\sum_{k=0}^{m-1}\ell(e_{m-k})\beta^{k}\right)\,\middle|\,e_{1}\ldots
e_{m}\text{ a path in }\mathsf{A}\right\\}.$

Since $\mu^{+}(A_{x})>0$ there is an $n\geq 1$ such that
$A_{x}\cap\sigma^{-n}(A_{x})\neq\emptyset$. This implies that

$x\in\left\\{\beta^{-n}\left(x+\ell(e_{n})+\cdots+\ell(e_{1})\beta^{n-1}\right)\,\middle|\,e_{1}\ldots
e_{n}\text{ a path in }\mathsf{A}\right\\},$

from which shows that $x\in\mathbb{Q}(\beta)$. Then there exists
$N\in\mathbb{N}$ such that $x\in\frac{1}{N}\mathbb{Z}[\beta]$.

Now fix $x\in P$ and and thus $N$. Let $y\in P$. Then there exists an $n\geq
1$ such that $A_{x}\cap\sigma^{-n}(A_{y})\neq\emptyset$ again by ergodicity.
This implies that there exists a path $e_{1}\ldots e_{m}$ in $\mathsf{A}$ such
that

$y=\beta^{n}x-\sum_{k=0}^{m-1}\ell(e_{m-k})\beta^{k},$

which shows that $y\in\frac{1}{N}\mathbb{Z}[\beta]$. Thus
$P\subset\frac{1}{N}\mathbb{Z}[\beta]$. Now by definition $P$ is bounded.
Applying the conjugations $\beta\mapsto\beta_{q}$ ($q=2,\ldots,r$) gives

$|y_{q}|\leq|\beta_{q}|^{n}|x_{q}|+\sum_{k=0}^{m-1}|\ell(e_{m-k})||\beta_{q}|^{k}\leq|x_{q}|+\frac{M}{1-|\beta_{q}|},$

if $|a|\leq M$ for all $a\in\mathcal{A}$. Thus $NP$ is a set of algebraic
integers all of whose conjugates are bounded. Thus $P$ is finite. ∎

######  Lemma 3.

Let $\beta$ be a Pisot number. Then the set of words

$\mathcal{L}_{0}=\left\\{(x_{1},x_{2},\ldots,x_{n})\in\mathcal{A}^{*}\,\middle|\,\sum_{k=1}^{n}\frac{x_{k}}{\beta^{k}}=0\right\\}$

is recognisable by a finite automaton. As a consequence the set

(15)
$\mathcal{K}_{0}^{+}=\left\\{(x_{1},x_{2},\ldots)\in\mathcal{A}^{\mathbb{N}}\,\middle|\,\sum_{k=1}^{\infty}\frac{x_{k}}{\beta^{k}}=0\right\\}$

is a space $\mathcal{K}_{I}^{+}$ for that automaton and an appropriate initial
state $I$ (labelled by $0$).

###### Proof.

Assume that $\mathcal{L}_{0}\neq\emptyset$; otherwise the assertion is
trivial. We define the set

$E=\left\\{\sum_{k=1}^{m-s}\frac{x_{k+s}}{\beta^{k}}\middle|0\leq
s<m,(x_{1},\ldots,x_{m})\in\mathcal{L}_{0}\right\\}.$

This will be the set of states. Between two elements $x,y\in E$ there is a
transition, if and only if

$a=\beta x-y\in\mathcal{A};$

this transition will then be marked with $a$. Since by definition of $E$ every
element of $E$ can be obtained from $0\in E$ by finitely many transitions
$x\mapsto\beta x-a$ with $a\in\mathcal{A}$. This shows that every element of
$E$ can be expressed in the form

$-\sum_{k=0}^{m}x_{k}\beta^{k}\quad\text{with
}x_{0},\ldots,x_{m}\in\mathcal{A}.$

This shows that all elements of $E$ are algebraic integers. The original
definition of $E$ shows that all elements of $E$ are bounded by
$\frac{M}{\beta-1}$, if $|a|\leq M$ for all $a\in\mathcal{A}$. Furthermore,
all conjugates of elements of $E$ are bounded by

$\left|-\sum_{k=0}^{m}x_{k}\beta_{q}^{k}\right|\leq\frac{M}{1-|\beta_{q}|}$

for $q=2,\ldots,r$. Thus the set $E$ consists of algebraic integers all of
whose conjugates are bounded; thus $E$ is finite. The set $E$ together with
the transitions defined above define a finite automaton recognising
$\mathcal{L}_{0}$ by taking $I=\\{0\\}$ as initial and terminal state.

All infinite words recognised by this automaton correspond to sums as in the
definition of $\mathcal{K}_{0}$. ∎

######  Lemma 4.

Let $x\in\mathbb{R}$ be such that $\nu(\\{x\\}))>0$. Then the set
$(\phi^{+})^{-1}(\\{x\\})$ is open.

###### Proof.

Since the set $P$ of atoms of $\nu$ is finite,
$(\phi^{+})^{-1}(\\{x\\})=(\phi^{+})^{-1}(x-\varepsilon,x+\varepsilon)$ for
small enough $\varepsilon>0$. The continuity of $\phi^{+}$ implies that
$(\phi^{+})^{-1}(\\{x\\})$ is open. ∎

######  Theorem 2.

The set $\phi^{+}(\mathcal{K}^{+})$ is either finite or perfect and thus
uncountable. In the first case the measure $\phi^{+}_{*}(\mu)$ is atomic, in
the second case it is continuous.

###### Proof.

The set $\phi^{+}(\mathcal{K}^{+})$ is compact as the continuous image of a
compact set.

Assume that there exists a vertex $v\in V$ and two paths $e_{1}e_{2}\ldots
e_{L_{1}}$ and $f_{1}f_{2}\ldots f_{L_{2}}$ both connecting $v$ to itself such
that

(16)
$\frac{1}{1-\beta^{-L_{1}}}\sum_{k=1}^{L_{1}}\frac{\ell(e_{k})}{\beta^{k}}\neq\frac{1}{1-\beta^{-L_{2}}}\sum_{k=1}^{L_{2}}\frac{\ell(f_{k})}{\beta^{k}}.$

Then we will show that the set $\phi^{+}(\mathcal{K}^{+})$ is perfect. For
this purpose we choose $x\in\phi^{+}(\mathcal{K}^{+})$ and show that there is
a sequence $(x_{n})_{n\in\mathbb{N}}$ of points in $\phi^{+}(\mathcal{K}^{+})$
with $x=\lim_{n\to\infty}x_{n}$ and $x_{n}\neq x$ for all $n$.

Let $x=\phi^{+}((\ell(a_{1}),\ell(a_{2}),\ldots))$. For $n\in\mathbb{N}$ we
set

$\displaystyle\xi_{n}$
$\displaystyle=\phi^{+}(\ell(a_{1}),\ell(a_{2}),\ldots,\ell(a_{n}),\ell(b_{1}),\ldots,\ell(b_{k}),\overline{\ell(e_{1}),\ldots,\ell(e_{L_{1}})})$
$\displaystyle\eta_{n}$
$\displaystyle=\phi^{+}(\ell(a_{1}),\ell(a_{2}),\ldots,\ell(a_{n}),\ell(b_{1}),\ldots,\ell(b_{k}),\overline{\ell(f_{1}),\ldots,\ell(f_{L_{2}})}),$

where $b_{1},\ldots,b_{k}\in E$ are chosen so that
$\ell(a_{1})\ldots\ell(a_{n})\ell(b_{1})\ldots\ell(b_{k})\in\mathcal{L}$ and
$\mathrm{t}(b_{k})=v$. There exists an integer $k\leq\\#V$ with this property
for every $n$. By (16) $\xi_{n}\neq\eta_{n}$, and thus at least one of these
two values is different from $x$. We take $x_{n}$ to be this value. Then
$\lim_{n\to\infty}x_{n}=x$ showing that $x$ is not isolated.

By Lemma 4 the preimage of any atom of the measure $\phi^{+}_{*}(\mu)$ would
contain a cylinder set. This would contradict the fact that we have just
proved that the images of all cylinder sets are uncountable. This shows that
$\nu$ is continuous in this case.

Assume on the contrary that for all $v\in V$ there exists a value
$c(v)\in\mathbb{R}$ such that for all paths $e_{1}e_{2}\ldots e_{n}$
connecting $v$ to itself

(17) $c(v)=\frac{1}{1-\beta^{-n}}\sum_{k=1}^{n}\frac{\ell(e_{k})}{\beta^{k}}.$

In this case every infinite path $e_{1}e_{2}\ldots$ starting at $v$ yields the
value $c(v)$ for $\phi^{+}$: assume that $j$ is chosen to be the minimal index
so that $w=\mathrm{t}(e_{j})$ is visited infinitely often by the path. If
$j=1$, the value of $\phi^{+}$ given by the path is $c(v)$ by definition. For
$j>1$ we decompose the path

$e_{1}\ldots e_{k_{1}}e_{k_{1}+1}\ldots e_{k_{2}}e_{k_{2}+1}\ldots
e_{k_{3}}\ldots,$

where $k_{1}=j$ and $\mathrm{t}(e_{k_{m}})=w$. Then by our assumption (17)
every path $e_{k_{m}+1}\ldots e_{k_{m+1}}$ can be replaced by a path
$f_{1}\ldots f_{s}f_{s+1}\ldots f_{s+q}$ with $\mathrm{i}(f_{1})=w$,
$\mathrm{t}(f_{s})=\mathrm{i}(f_{s+1})=v$, and $\mathrm{t}(f_{s+q})=w$ without
changing the value of $\phi^{+}$. The new path visits $v$ infinitely often,
and thus assigns the value $c(v)$ to $\phi^{+}$. Thus $\phi^{+}$ only takes
the values $\\{c(v)\mid v\in V\\}$. Each of these values is assigned a
positive mass. This proves that the number of atoms of $\phi^{+}_{*}(\mu)$ is
bounded by the number of vertices of $\mathsf{A}$. This shows the last
assertion of Theorem 1. ∎

The following result is a consequence of the proof of Theorem 2.

######  Theorem 3.

The set $\phi^{+}(\mathcal{K}^{+})$ is finite, if and only if for every vertex
$v\in V$ there exists a value $c(v)\in\mathbb{R}$ such that for all paths
$e_{1}e_{2}\ldots e_{n}$ connecting $v$ to itself (17) holds.

######  Remark 1.

Notice that the measure $\phi_{*}^{+}(\mu)$ can only be absolutely continuous,
if $\beta\leq\lambda$ (recall that $\lambda$ is the Perron-Frobenius
eigenvalue associated to the automaton $\mathsf{A}$). Otherwise the set
$\phi^{+}(\mathcal{K}^{+})$ has Hausdorff dimension
$\leq\frac{\log\lambda}{\log\beta}<1$ and cannot support an absolutely
continuous measure.

##  6\. Fourier transforms and matrix products

The Fourier transforms of the measures $\nu$ and $\nu_{I}$ are given by

(18) $\begin{split}\widehat{\nu}_{I}(t)=\int\limits_{-\infty}^{\infty}e^{-2\pi
ixt}\,d\nu_{I}(x)\\\ \widehat{\nu}(t)=\int\limits_{-\infty}^{\infty}e^{-2\pi
ixt}\,d\nu(x).\end{split}$

In order to derive expressions for $\widehat{\nu}_{I}$ and $\widehat{\nu}$, we
introduce the weighted transition matrix for $\phi^{+}$ and the underlying
automaton $\mathsf{A}$

(19) $W(t)=\frac{1}{\lambda}\sum_{a\in\mathcal{A}}e(-at)M_{a},$

where we use the notation $e(x)=e^{2\pi ix}$.

######  Proposition 1.

The Fourier transforms of the measures $\nu$ and $\nu_{I}$ can be expressed as

(20)
$\widehat{\nu}(t)=\mathbf{v}_{L}^{\mathsf{T}}\prod_{n=1}^{\infty}W(\beta^{-n}t)\mathbf{v}_{R}$

and

(21)
$\widehat{\nu}_{I}(t)=\frac{1}{\mathbf{v}_{I}^{\mathsf{T}}\mathbf{v}_{R}}\mathbf{v}_{I}^{\mathsf{T}}\prod_{n=1}^{\infty}W(\beta^{-n}t)\mathbf{v}_{R},$

where the infinite matrix product is interpreted so that the factors are
ordered from left to right.

###### Proof.

We set

$\phi_{n}^{+}((x_{k})_{k\in\mathbb{N}})=\sum_{k=1}^{n}\frac{x_{k}}{\beta^{k}}.$

Then
$\lim_{n\to\infty}\phi_{n}^{+}((x_{k})_{k\in\mathbb{N}})=\phi^{+}((x_{k})_{k\in\mathbb{N}})$
holds uniformly on $\mathcal{K}^{+}$. We set
$\nu_{n}=(\phi_{n}^{+})_{*}(\mu^{+})$ and observe that
$\nu_{n}\rightharpoonup\nu$. Then

$\widehat{\nu}_{n}(t)=\mathbf{v}_{L}^{\mathsf{T}}\prod_{k=1}^{n}W(\beta^{-k}t)\mathbf{v}_{R}.$

The limit relation $\lim_{n\to\infty}\widehat{\nu}_{n}=\widehat{\nu}$ and
equation (20) then follow by weak convergence. A similar reasoning gives (21).
∎

######  Remark 2.

As pointed out in Section 4 Erdős [13, 12] proved the singularity of the
distribution measure $\nu$ of the random series

$\sum_{n=1}^{\infty}\frac{X_{n}}{\beta^{n}},$

by showing that $(\widehat{\nu}(\beta^{k}))_{k\in\mathbb{N}}$ does not tend to
$0$. Of course the fact that $\lim_{|t|\to\infty}\widehat{\nu}(t)=0$ does not
suffice in general to prove absolute continuity, as there are singular
measures, so called Rajchman measures, whose Fourier transform vanishes at
$\infty$ (see [31]).

Using the Pisot property of $\beta$ we define the map

(22)
$\Phi:\mathcal{K}\to\mathbb{T}^{r},(x_{k})_{k\in\mathbb{Z}}\mapsto\left(\sum_{k=-\infty}^{\infty}x_{k}\beta^{-k+m}\pmod{1}\right)_{m=0}^{r-1},$

where we use the notation $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. Notice that the
series for $k\leq 0$ converges $\pmod{1}$ by the fact that

$\beta^{m}+\beta_{2}^{m}+\cdots+\beta_{r}^{m}\in\mathbb{Z}$

and $|\beta_{2}|,\ldots,|\beta_{r}|<1$.

######  Remark 3.

A similar map was studied in [39, 38]. In [38] this map was used to give a
dynamic proof of the singularity of the Erdős measure introduced in [12]. In
[39] conditions were given under which this map between the two-sided
$\beta$-shift $X_{\beta}$ and $\mathbb{T}^{d}$ is almost bijective. In the
present situation, where $\beta$ and the underlying language $\mathcal{L}$ are
not necessarily related, nothing can be said about (almost) bijectivity of
this map.

######  Theorem 4.

Let $z=m_{0}+m_{1}\beta+\cdots+m_{r-1}\beta^{r-1}\in\mathbb{Z}[\beta]$ for
$\beta$ a Pisot number of degree $r$ and $\nu$ be the measure given by (14).
Then the limit

(23)
$\widehat{\psi}(m_{0},\ldots,m_{r-1})=\lim_{k\to\infty}\widehat{\nu}(z\beta^{k})$

exists. These values are the Fourier coefficients of the measure
$\psi=\Phi_{*}(\mu)$ on $\mathbb{T}^{r}$.

###### Proof.

We define the maps

$\Phi_{n}((x_{k})_{k\in\mathbb{Z}})=\left(\sum_{k=-n}^{\infty}x_{k}\beta^{-k+m}\pmod{1}\right)_{m=0}^{r-1}.$

Then $(\Phi_{n})_{n\in\mathbb{N}}$ converges to $\Phi$ uniformly on
$\mathcal{K}$. Let $\psi_{n}=(\Phi_{n})_{*}(\mu)$. Then
$\psi_{n}\rightharpoonup\psi$ and

$\widehat{\psi}_{n}(m_{0},\ldots,m_{r-1})=\widehat{\nu}(z\beta^{n}).$

The limit relation (23) then follows by weak convergence and the fact that
$(\beta^{n}\mod 1)\to 0$. ∎

######  Remark 4.

The shift on $\mathcal{K}$ is conjugate via $\Phi$ to the hyperbolic toral
endomorphism

$B:\mathbb{T}^{r}\to\mathbb{T}^{r},\begin{pmatrix}t_{0}\\\ t_{1}\\\ \vdots\\\
t_{r-1}\end{pmatrix}\mapsto\begin{pmatrix}t_{1}\\\ \vdots\\\ t_{r-1}\\\
a_{0}t_{0}+\cdots+a_{r-1}t_{r-1}\pmod{1}\end{pmatrix},$

where

$\beta^{r}=a_{r-1}\beta^{r-1}+\cdots+a_{1}\beta+a_{0}$

is the minimal equation of $\beta$. The measure $\psi$ is then a $B$-invariant
measure on $\mathbb{T}^{r}$, and $B$ is ergodic with respect to $\psi$.

######  Theorem 5.

The measure $\nu$ given by (14) is absolutely continuous, if and only if for
all $z\in\mathbb{Z}[\beta]\setminus\\{0\\}$

(24) $\lim_{k\to\infty}\widehat{\nu}(z\beta^{k})=0.$

###### Proof.

Let $\beta_{2},\ldots,\beta_{r}$ denote the Galois conjugates of $\beta$ and
assume that $\beta,\beta_{2},\ldots,\beta_{s}\in\mathbb{R}$ and
$\beta_{s+1},\beta_{s+2},\ldots,\beta_{s+t},\overline{\beta_{s+1}},\overline{\beta_{s+2}},\ldots,\overline{\beta_{s+t}}\in\mathbb{C}\setminus\mathbb{R}$;
then $r=s+2t$. Then the map

$\widetilde{\Phi}:\mathcal{K}\to\mathbb{R}^{s}\times\mathbb{C}^{t},(x_{k})_{k\in\mathbb{Z}}\mapsto\left(\sum_{k=1}^{\infty}x_{k}\beta^{-k},\sum_{k=0}^{\infty}x_{-k}\beta_{2}^{k},\ldots,\sum_{k=0}^{\infty}x_{-k}\beta_{s+t}^{k}\right)$

is continuous and $\widetilde{\Phi}(\mathcal{K})$ is compact. Together with
the map

$\rho:\mathbb{R}^{s}\times\mathbb{C}^{t}\to\mathbb{R}^{r},(y_{1},\ldots,y_{s},z_{s+1},\ldots,z_{s+t})\mapsto\\\
\left(\beta^{m}y_{1}-(\beta_{2}^{m}y_{2}+\cdots+\beta_{s}^{m}y_{s})-2\Re(\beta_{s+1}^{m}z_{s+1}+\cdots+\beta_{s+t}^{m}z_{s+t})\right)_{m=0}^{r-1}$

we have

$\Phi=\rho\circ\widetilde{\Phi}\pmod{1}.$

The map $\rho$ is linear and $\rho\pmod{1}$ is finite to one on
$\widetilde{\Phi}(\mathcal{K})$. Now the measure
$\psi=\\!\\!\pmod{1}_{*}\circ\rho_{*}\circ\widetilde{\Phi}_{*}(\mu)$ is equal
to the Lebesgue measure, if and only if
$\widehat{\psi}(m_{0},\ldots,m_{r-1})=0$ for all
$(m_{0},\ldots,m_{r-1})\in\mathbb{Z}^{r}\setminus\\{\mathbf{0}\\}$. If $\psi$
is Lebesgue measure, then the measure
$\nu=P_{*}\circ\widetilde{\Phi}_{*}(\mu)$ is absolutely continuous, where $P$
denotes the projection to the first coordinate. On the other hand, if $\psi$
is not the Lebesgue measure, then $\nu$ cannot be absolutely continuous by
(23). ∎

##  7\. Examples

######  Example 1.

Let $\mathcal{K}\subset\\{0,\pm 1\\}^{\mathbb{N}}$ be given by the automaton
in Figure 1 and take $\beta=\frac{1+\sqrt{5}}{2}$. Then the map $\phi^{+}$
takes the values

$\displaystyle 0$ $\displaystyle\nu(\\{0\\})=\frac{1}{\gamma^{2}}$
$\displaystyle\pm$ $\displaystyle 1$
$\displaystyle\nu(\\{1\\})=\nu(\\{-1\\})=\frac{1}{2}\left(\frac{1}{\gamma}-\frac{1}{\gamma^{2}}\right)$
$\displaystyle\pm$ $\displaystyle\frac{1}{\beta}$
$\displaystyle\nu\left(\left\\{\frac{1}{\beta}\right\\}\right)=\nu\left(\left\\{-\frac{1}{\beta}\right\\}\right)=\frac{1}{\gamma^{3}}$

with the indicated probabilities, where $\gamma$ is the positive solution of

$x^{3}=x^{2}+2.$

The value $\gamma$ is the Perron-Frobenius eigenvalue of the adjacency matrix
of the automaton given by Figure 1.

Figure 1. The automaton recognising all expansions of $0$ in base
$\frac{1+\sqrt{5}}{2}$ with digits $\\{0,\pm 1\\}$

######  Example 2.

Let $\mathcal{K}\subset\\{0,1\\}^{\mathbb{N}}$ be the set of sequences of $0$
and $1$ with no two consecutive $1$s (given by the automaton in Figure 2).
These are the digital representations of all real numbers in $[0,1]$ obtained
by iteration of the $\beta$-transformation $T_{\beta}:x\mapsto\beta x\mod 1$
for $\beta=\frac{1+\sqrt{5}}{2}$ (see [26]). Then the measure $\phi^{+}(\mu)$
is Lebesgue measure on $[0,1]$.

Figure 2. The automaton recognising all greedy expansions in base
$\frac{1+\sqrt{5}}{2}$ with digits $\\{0,1\\}$

######  Example 3.

Take $\mathcal{K}=\\{0,1,2,3\\}^{\mathbb{N}}$ and $\mu$ the infinite product
measure assigning probability $\frac{1}{4}$ to each letter. Take $\beta=2$ and
consider the map $\phi^{+}$ as above. Then the corresponding measure on
$[0,3]$ is given by the density

$h(x)=\begin{cases}\frac{1}{2}x&\text{for }0\leq x\leq 1\\\
\frac{1}{2}&\text{for }1\leq x\leq 2\\\ \frac{1}{2}(3-x)&\text{for }2\leq
x\leq 3\\\ 0&\text{otherwise.}\end{cases}$

Projecting this measure $\mod 1$ gives Lebesgue measure, because in this case
$r=1$ and the map $\Phi$ maps to $\mathbb{T}$.

######  Example 4.

Let $\mathcal{K}^{+}\subset\\{0,1,2\\}^{\mathbb{N}}$ be the set of sequences
recognised by the automaton in Figure 3. These are all expansions of real
numbers expressed in base $\beta^{2}$, where $\beta=\frac{1+\sqrt{5}}{2}$.
This is similar to Example 3, where expansions in base $\beta^{2}$ are
interpreted in base $\beta$. In this case the measure is singular, though.
This can be seen by computing

$\lim_{n\to\infty}\widehat{\nu}(\beta^{n})=0.0608424\ldots+0.0208583\ldots i.$

numerically.

Figure 3. The automaton recognising all expansions in base
$\beta^{2}=\frac{3+\sqrt{5}}{2}$ with digits $\\{0,1,2\\}$

######  Example 5.

This example is taken from [17]. Take $\mathcal{K}\subset\\{0,\pm
1\\}^{\mathbb{N}}$ to be the set of sequences recognised by the automaton in
Figure 4. The automaton with initial state $I$ recognises all expansions of
integers in the form

$n=\sum_{k=0}^{K}x_{k}2^{k}\quad\text{with }x_{k}\in\\{0,\pm 1\\}$

which minimise the weight

$w(n)=\sum_{k=0}^{K}|x_{k}|.$

This is motivated by the fact that this weight is the number of
additions/subtractions in computing $nP$ by a Horner-type scheme, where $P$ is
a point on an elliptic curve. The measures $\phi_{*}^{+}(\mu^{+})$ and
$\phi_{*}^{+}(\mu_{I}^{+})$ are singular in this case, as was proved in [17].

Figure 4. The automaton recognising expansions in base $2$ with digits
$\\{0,\pm 1\\}$

######  Acknowledgement.

The author is indebted to Jörg Thuswaldner for very valuable discussions on
the subject (several years ago in a café in Pisa).

## References

  * [1] M. Baake and U. Grimm, _Aperiodic order. Vol. 1_ , Encyclopedia of Mathematics and its Applications, vol. 149, Cambridge University Press, Cambridge, 2013, A mathematical invitation, With a foreword by Roger Penrose.
  * [2] by same author, _Fourier transform of Rauzy fractals and point spectrum of 1D Pisot inflation tilings_ , arXiv:1907.11012, July 2019.
  * [3] M.-P. Béal and D. Perrin, _Symbolic dynamics and finite automata_ , Handbook of formal languages, Vol. 2, Springer, Berlin, 1997, pp. 463–505.
  * [4] M.-J. Bertin, A. Decomps-Guilloux, M. Grandet-Hugot, M. Pathiaux-Delefosse, and J.-P. Schreiber, _Pisot and Salem numbers_ , Birkhäuser Verlag, Basel, 1992, With a preface by David W. Boyd.
  * [5] J. Brémont, _Self-similar measures and the Rajchman property_ , Ann. H. Lebesgue (2021), to appear, https://arxiv.org/abs/1910.03463.
  * [6] E. Breuillard and P. P. Varjú, _On the dimension of Bernoulli convolutions_ , Ann. Probab. 47 (2019), no. 4, 2582–2617.
  * [7] I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ, _Ergodic theory_ , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 245, Springer-Verlag, New York, 1982, Translated from the Russian by A. B. Sosinskiĭ.
  * [8] K. Dajani and C. Kraaikamp, _Ergodic theory of numbers_ , Carus Mathematical Monographs, vol. 29, Mathematical Association of America, Washington, DC, 2002.
  * [9] R. Diestel, _Graph theory_ , fifth ed., Graduate Texts in Mathematics, vol. 173, Springer, Berlin, 2017.
  * [10] S. Eilenberg, _Automata, languages, and machines. Vol. A_ , Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York, 1974, Pure and Applied Mathematics, Vol. 58.
  * [11] P. D. T. A. Elliott, _Probabilistic number theory. I, mean-value theorems_ , Grundlehren der Mathematischen Wissenschaften, vol. 239, Springer-Verlag, New York, 1979.
  * [12] P. Erdős, _On a family of symmetric Bernoulli convolutions_ , Amer. J. Math. 61 (1939), 974–976.
  * [13] by same author, _On the smoothness properties of a family of Bernoulli convolutions_ , Amer. J. Math. 62 (1940), 180–186.
  * [14] D.-J. Feng and H. Hu, _Dimension theory of iterated function systems_ , Comm. Pure Appl. Math. 62 (2009), no. 11, 1435–1500.
  * [15] C. Frougny, _Representations of numbers and finite automata_ , Math. Systems Theory 25 (1992), no. 1, 37–60.
  * [16] by same author, _Numeration systems_ , ch. 7, pp. 230–268, vol. 90 of Encyclopedia of Mathematics and its Applications [25], 2002.
  * [17] P. J. Grabner and C. Heuberger, _On the number of optimal base 2 representations of integers_ , Des. Codes Cryptogr. 40 (2006), 25–39.
  * [18] P. J. Grabner, C. Heuberger, and H. Prodinger, _Distribution results for low-weight binary representations for pairs of integers_ , Theor. Comput. Sci. 319 (2004), 307–331.
  * [19] by same author, _Counting optimal joint digit expansions_ , Integers 5 (2005), no. 3, A09, 19 pages (electronic).
  * [20] P. J. Grabner, P. Liardet, and R. F. Tichy, _Spectral disjointness of dynamical systems related to some arithmetic functions_ , Publ. Math. Debrecen 66 (2005), 213–244.
  * [21] P. J. Grabner and W. Steiner, _Redundancy of minimal weight expansions in Pisot bases_ , Theor. Comput. Sci 412 (2011), 6303–6315.
  * [22] M. A. Harrison, _Introduction to formal language theory_ , Addison-Wesley Publishing Co., Reading, Mass., 1978.
  * [23] B. Jessen and A. Wintner, _Distribution functions and the Riemann zeta function_ , Trans. Amer. Math. Soc. 38 (1935), 48–88.
  * [24] D. Lind and B. Marcus, _An introduction to symbolic dynamics and coding_ , Cambridge University Press, Cambridge, 1995.
  * [25] M. Lothaire, _Algebraic combinatorics on words_ , Encyclopedia of Mathematics and its Applications, vol. 90, Cambridge University Press, Cambridge, 2002.
  * [26] W. Parry, _On the $\beta$-expansions of real numbers_, Acta Math. Acad. Sci. Hung. 11 (1960), 401–416.
  * [27] W. Parry, _Intrinsic Markov chains_ , Trans. Amer. Math. Soc. 112 (1964), 55–66.
  * [28] by same author, _Symbolic dynamics and transformations of the unit interval_ , Trans. Amer. Math. Soc. 122 (1966), 368–378.
  * [29] Y. Peres, W. Schlag, and B. Solomyak, _Sixty years of Bernoulli convolutions_ , Fractal geometry and stochastics, II (Greifswald/Koserow, 1998), Progr. Probab., vol. 46, Birkhäuser, Basel, 2000, pp. 39–65.
  * [30] M. Queffélec, _Substitution dynamical systems—spectral analysis_ , second ed., Lecture Notes in Mathematics, vol. 1294, Springer-Verlag, Berlin, 2010\.
  * [31] A. Rajchman, _Une classe de séries trigonométriques qui convergent presque partout vers zéro_ , Math. Ann. 101 (1929), no. 1, 686–700.
  * [32] A. Rényi, _Representation for real numbers and their ergodic properties_ , Acta Math. Acad. Sci. Hung. 8 (1957), 477–493.
  * [33] S. Saglietti, P. Shmerkin, and B. Solomyak, _Absolute continuity of non-homogeneous self-similar measures_ , Adv. Math. 335 (2018), 60–110.
  * [34] J. Sakarovitch, _Elements of automata theory_ , Cambridge University Press, Cambridge, 2009, Translated from the 2003 French original by Reuben Thomas.
  * [35] E. Seneta, _Nonnegative matrices and Markov chains_ , second ed., Springer Series in Statistics, Springer-Verlag, New York, 1981.
  * [36] P. Shmerkin, _On the exceptional set for absolute continuity of Bernoulli convolutions_ , Geom. Funct. Anal. 24 (2014), no. 3, 946–958.
  * [37] P. Shmerkin and B. Solomyak, _Absolute continuity of self-similar measures, their projections and convolutions_ , Trans. Amer. Math. Soc. 368 (2016), no. 7, 5125–5151.
  * [38] N. Sidorov and A. Vershik, _Ergodic properties of the Erdős measure, the entropy of the golden shift, and related problems_ , Monatsh. Math. 126 (1998), no. 3, 215–261.
  * [39] N. A. Sidorov, _Bijective and general arithmetic codings for Pisot toral automorphisms_ , J. Dynam. Control Systems 7 (2001), no. 4, 447–472.
  * [40] B. Solomyak, _On the random series $\sum\pm\lambda^{n}$ (an Erdős problem)_, Ann. of Math. (2) 142 (1995), no. 3, 611–625.
  * [41] P. P. Varjú, _Absolute continuity of Bernoulli convolutions for algebraic parameters_ , J. Amer. Math. Soc. 32 (2019), no. 2, 351–397.
  * [42] by same author, _On the dimension of Bernoulli convolutions for all transcendental parameters_ , Ann. of Math. (2) 189 (2019), no. 3, 1001–1011.
  * [43] Péter P. Varjú, _Recent progress on Bernoulli convolutions_ , European Congress of Mathematics, Eur. Math. Soc., Zürich, 2018, pp. 847–867.
  * [44] P. Walters, _Ergodic theory_ , Springer, Berlin, 1982.
  * [45] R. Winkler, _The order theoretic structure of the set of $P$-sums of a sequence_, Publ. Math. Debrecen 58 (2001), no. 3, 467–490.