2.1 $\beta $ -shifts

Recall our notation $\Lambda _{b}=\{ 0,1,\ldots ,[b]-1\} $ . With respect to the product $\sigma $ -algebra on $\Lambda _{b}^{\mathbb {N}}$ , the shift transformation $\sigma :\Lambda _{b}^{\mathbb {N}}\rightarrow \Lambda _{b}^{\mathbb {N}}$ which is defined by $\sigma ((\unicode{x3bb} _{i}))=(\unicode{x3bb} _{i+1})$ turns $\Lambda _{b}^{\mathbb {N}}$ into a dynamical system that we call the full $[b]$ shift. The restriction of the full shift to the closure of the subset of sequences which encodes b-expansions is a subshift that we call the b-shift and denote by $X_{b}\subset \Lambda _{b}^{\mathbb {N}}$ .

One says that a real positive b is the following.

• • A simple number if the b-expansion of $1$ has a $0$ s tail.

• • A simple Parry number if the b-expansion of $1$ is a periodic sequence. In some sources, it is also called a (purely) periodic number.

• • A Parry number if the b-expansion of $1$ has a periodic tail. In some sources, it is also called an eventually periodic number.

It is immediate to conclude that

$$ \begin{align*} \{ \text{simple } \#\text{s} \} \subset \{ \text{simple Parry } \#\text{s} \} \subset \{ \text{Parry } \#\text{s} \} \subset \{ \text{specified } \#\text{s} \}. \end{align*} $$

Parry showed [Reference Parry9] that the simple numbers are everywhere dense in $(1,\infty )$ , and so is the set of specified numbers. Schmeling [Reference Schmeling11] showed that the set of specified numbers also has Hausdorff dimension $1$ , but it is meager and has Lebesgue measure $0$ . In particular, it has the cardinality of the continuum.

An important property of specified numbers is the following proposition.

Finally, we present a result by Parry [Reference Parry9] that gives a criterion to determine whether a given sequence $(b_{n})\in \{ 0,\ldots [b]\} ^{\mathbb {N}}$ is a b-expansion of some $x\in [0,1)$ , $x=b_{0}+{b_{1}}/{\beta }+\cdots $ . We emphasize that there might be many representations of x in this form but only one of them corresponds to the b-expansion that we described earlier. This will be useful for constructing the counterexample in §7.

If $(a_{0},a_{1},\ldots ),(b_{0},b_{1},\ldots )$ are sequences of the same length (finite or infinite) of non-negative integers less than b, we write $(a_{0},a_{1},\ldots )<(b_{0},b_{1},\ldots )$ when $a_{n}<b_{n}$ for the first $a_{n}\neq b_{n}$ .

2.2 Entropy theory

Let $(X,\mathcal {B})$ be a standard Borel space and $\mathcal {D}\subset \mathcal {B}$ be a measurable partition. We write $\mathcal {\mathcal {D}}(x)\in \mathcal {\mathcal {D}}$ for the element which contains $x\in X$ . This is also well defined when $\mathcal {\mathcal {D}}$ is countably generated $\sigma $ -algebra. In addition, we denote the joining of two finite partitions $\mathcal {A},\mathcal {B}$ by $\mathcal {A\lor \mathcal {B}}=\{ A\cap B:A\in \mathcal {A},B\in \mathcal {B}\} $ . Let $(X,\mathcal {B},\mu )$ be a probability space then the Shannon entropy of $\mu $ with respect to a partition $\mathcal {A}$ of X is the non-negative number $H_{\mu }(A)=-\sum _{A\in \mathcal {A}}\mu (A)\log \mu (A)$ . The entropy $h_{\mu }(T,\mathcal {A})$ of a partition $\mathcal {A}$ of a measure-preserving system $(X,\mathcal {F},\mu ,T)$ is the limit $h_{\mu }(T,\mathcal {A})=\lim _{n\rightarrow \infty }({1}/{n})H_{\mu }(\mathcal {A}_{n})$ , where $\mathcal {A}_{n}={ \bigvee_{i=0}^{n-1}}T^{-i}\mathcal {A}$ . The Kolmagorov–Sinai entropy (or just the entropy) of the m.p.t. $(X,\mathcal {F},\mu ,T)$ is $h_{\mu }(T)=\sup _{\mathcal {A}}h_{\mu }(T,\mathcal {A})$ where the supremum is taken over all the finite partitions $\mathcal {A}$ . Equality is achieved if $\mathcal {A}$ is a generating partition, $\mathcal {F}={ \bigvee _{n=1}^{\infty }}\mathcal {A}_{n}\,\mod \mu $ .

A landmark result in ergodic theory is the Shannon–McMillan–Breiman theorem [Reference Walters14]. Let $(X,\mathcal {F},\mu ,T)$ be an ergodic measure-preserving system and $\mathcal {A}$ a finite partition. Then $\mu $ -a.e. $x \lim _{n\rightarrow \infty }({1}/{n})\log \mu (\mathcal {A}_{n}(x))=h_{\mu }(T,\mathcal {A})$ . It is not hard to deduce from it that an ergodic and $T_{a}$ -invariant Borel probability measure $\mu $ with positive entropy is non-atomic (this also can be proved directly). We will use this corollary in occasional places.

2.3 General results on equidistribution (due to Hochman)

This subsection covers three results that we adopt from Hochman [Reference Hochman4]. Two of them are presented with a very superficial description of their proofs to help the reader gain some intuition. A more thorough treatment can be found in the original paper.

We denote the real line translation and scaling maps by

$$ \begin{align*} R_{\theta}x & =x+\theta,\\ S_{t}x & =t\cdot x, \end{align*} $$

respectively. Here $R_{\theta }$ is taken $\mod 1$ when acting on $[0,1)\cong \mathbb {R}/\mathbb {Z}$

Let $\mu $ be a probability measure on $\mathbb {R}/\mathbb {Z}$ and $E\in \mathcal {B}$ such that $\mu (E)>0$ . We write $\mu _{E}=({1}/{\mu (E)})\cdot \mu |_{E}$ for the normalized restriction of $\mu $ to E.

The next technique relates orbits to the local structure of $\mu $ .

Theorem 2.3. [Reference Hochman4]

Let $T:X\rightarrow X$ be a continuous map of compact metric space. Let $\mathcal {D}_{1},\mathcal {D}_{2},\ldots $ be a refining sequence of finite Borel partitions. Let $\mu $ be a Borel probability measure on X and assume that $\sup _{n\in \mathbb {N}}\{ \mathop {}\!\mathrm {diam} T_{a}^{n}D:D\in \mathcal {D}_{n+k},\mu (D)>0\} \rightarrow 0$ as $k\rightarrow \infty $ . Then for $\mu $ -a.e. x,

$$ \begin{align*} \bigg(\frac{1}{N}\sum_{n=1}^{N}\delta_{T^{n}x}-\frac{1}{N}\sum_{n=1}^{N}T^{n}\mu_{\mathcal{D}_{n} (x )} \bigg)\xrightarrow[w-*]{N\rightarrow\infty}0. \end{align*} $$

The idea of the proof is to take a countable dense set in $C(X)$ and prove the weak-* convergence with respect to its members. The left average can be replaced with the $\mathcal {D}_{n+k}$ -conditional mean by the assumption. The right average is just the $\mathcal {D}_{n}(x)$ -conditional mean. A variant of the ergodic theorem for martingale differences implies that their limits are equal.

The second theorem is about equidistribution along orbits of the form $(n\theta ,T^{[\beta n]}x)$ , where x is a typical point for $\mu $ .

The proof uses an intermediate result which says that for $\tau \times \mu $ -a.e. $(u,x)$ , the orbit $(n\theta +u,T^{[\beta n]}x)$ equidistributes for a measure $\nu _{u,x}$ on $[0,1)\times X$ satisfying

$$ \begin{align*} \intop\!\nu_{u,x}\,dz\times d\mu (u,x )=\tau\times\mu. \end{align*} $$

We will not go into its detail besides mentioning that a suspension of height $1$ is used to overcome the integer part issue. Back to the proof of Theorem 2.4, it implies that Lebesgue-a.e. $u\in [0,1)$ and $\mu $ -a.e. x, $\delta _{n\theta +u}\times \delta _{T^{[n\beta ]}x}\xrightarrow {w-*}\nu _{u,x}$ and also that $\intop \!\nu _{u,x}\,dz\times \mu (u,x)=m\times \mu $ . By translation of the first coordinate, we get that $(n\theta ,T^{[\beta n]}x)$ equidistributes for a measure $\nu _{x}$ on $[0,1)\times X$ , where $\nu _{x}=\nu _{x,0}$ , and with Lemma 2.3 from the original paper, we find that $\intop \!\nu _{x}\,d\mu (x)=\tau \times \mu $ .

Lastly, we give a slightly modified version for Hochman’s evaluation of the Fourier transform of scaled measures.

Lemma 2.5. Let $\mu $ be a non-atomic probability measure on ${\mathbb {R}}$ . Then for every $(c,d)\subset {\mathbb {R}}$ and for every $r>0$ and $m\neq 0$ ,

$$ \begin{align*} \intop\nolimits_{0}^{1} |\mathcal{F} ( (S_{b^{z}}\mu )|_{ [c,d ]} ) (m ) |^{2}\,dz\leq\frac{2\mu ( [c,d ] )^{2}}{r\cdot |m |}+\intop\nolimits_{c}^{d}\intop\nolimits_{c}^{d}\chi_{B_{r} (y' )} (y )\,d\mu (y )\,d\mu (y' ), \end{align*} $$

where $B_{r}(x)=\{ y: |x-y |<r\} $ .

Proof. (Based on Hochman’s proof [Reference Hochman4]) Using Fubini,

$$ \begin{align*} \intop\nolimits_{0}^{1} |\mathcal{F} ( (S_{b^{z}}\mu )|_{ [c,d ]} ) (m ) |^{2}\,dz & =\intop\nolimits_{0}^{1} \bigg|\intop\nolimits_{c}^{d}e (mb^{z}y )\,d\mu (y ) \bigg|^{2}\,dz\\ & =\intop\nolimits_{0}^{1}\intop\nolimits_{c}^{d}\intop\nolimits_{c}^{d}e (mb^{z}y )\overline{e (mb^{z}y' )}\,d\mu (y )\,d\mu (y' )\,dz\\ & =\intop\nolimits_{0}^{1}\intop\nolimits_{c}^{d}\intop\nolimits_{c}^{d}e (mb^{z} (y-y' ) )\,d\mu (y )\,d\mu (y' )\,dz\\ & =\intop\nolimits_{c}^{d}\intop\nolimits_{c}^{d}\intop\nolimits_{0}^{1}e (mb^{z} (y-y' ) )\,dz\,d\mu (y )\,d\mu (y' ) \end{align*} $$

and then changing of variables $t=b^{z}$ ,

$$ \begin{gather*} \leq\intop\nolimits_{c}^{d} \bigg(\!\intop\nolimits_{ [c,d ]\backslash B_{r} (y' )} \bigg|\intop\nolimits_{1}^{b}\frac{1}{\log (b )t}e (m (y-y' )t )\,dt \bigg|\,d\mu (y )+\intop\nolimits_{B_{r} (y' )\cap [c,d ]}1\,d\mu (y ) \bigg)\,d\mu (y' ). \end{gather*} $$

Finally, using integration by parts for the inner integral in the left summand,

$$ \begin{align*} \leq\frac{2\mu ( [c,d ] )^{2}}{r\cdot |m |}+\intop\nolimits_{c}^{d}\intop\nolimits_{c}^{d}\chi_{B_{r} (y' )} (y )\,d\mu (y )\,d\mu (y' ).\\[-48pt] \end{align*} $$

4.1 Relating the decay rate of a stochastic process to the decay rate of its mean

Let $(X,\mathcal {B},\mu )$ be a probability space and $\{ A_{m}\}_{m=1}^{\infty }$ be a stochastic process. A conventional application of Markov’s inequality and the first Borel–Cantelli lemma implies that the decay rate in m of $\{|A_{m}|\}_{m=1}^{\infty }$ relates to the decay rate in m of its means $\{\textbf {E}_{\mu }(|A_{m}|)\}_{m=1}^{\infty }$ in the following way.

Proof. Fix $c>1$ . By Markov’s inequality $\textbf {P}_{\mu }\{ |A_{m} |>({1}/{c})m^{-\alpha +\epsilon }\} <{c\textbf {E}_{\mu }|A_{m}|}/{m^{-\alpha +\epsilon }}$ and from the hypothesis above, there exists some $s>0$ such that $\textbf {P}_{\mu }\{ |A_{m} |>({1}/{c})m^{-\alpha +\epsilon }\} <{(c\cdot s\cdot m^{-1-\alpha })}/{m^{-\alpha +\epsilon }}$ . Thus,

$$ \begin{align*} &\textbf{P}_{\mu} \bigg\{ |A_{m} |>\frac{1}{c}m^{-\alpha+\epsilon} \bigg\} <\frac{c\cdot s\cdot m^{-1-\alpha}}{m^{-\alpha+\epsilon}}=c\cdot s\cdot m^{-1-\epsilon}\\ &\quad\implies\sum_{m}\textbf{P}_{\mu} \bigg\{ |A_{m} |>\frac{1}{c}m^{-\alpha+\epsilon} \bigg\} <\infty. \end{align*} $$

Finally, by the first Borel–Cantelli lemma, $\textbf {P}_{\mu }(\limsup _{m}\{|A_{m}|>({1}/{c})m^{-\alpha +\epsilon }\})=0$ , which implies that $\{|A_{m}|>({1}/{c})m^{-\alpha +\epsilon }\}$ occur finitely often with probability $1$ .

6.1 Stationary coding

The content of this subsection is taken from the book of P. Shields on ergodic theory [Reference Shields12].

Let A and B be finite sets. A Borel measurable map $F:A^{\mathbb {Z}}\rightarrow B^{\mathbb {Z}}$ is a stationary coder if $F(T_{A}x)=T_{B}F(x)$ for every $x\in A^{\mathbb {Z}}$ , where $T_{A}$ and $T_{B}$ denote the shifts on the respective two-sided sequence spaces $A^{\mathbb {Z}}$ and $B^{\mathbb {Z}}$ . Stationary coding carries a Borel measure $\mu $ on $A^{\mathbb {Z}}$ into the measure $\nu =\mu \circ F^{-1}$ defined for Borel subsets C of $B^{\mathbb {Z}}$ by $\nu (C)=\mu (F^{-1}(C))$ . The encoded process $\nu $ is said to be a stationary coding of $\mu $ , with coder F. It is immediate to conclude that a stationary coding of a stationary process is itself a stationary process.

The map $f:A^{\mathbb {Z}}\rightarrow B$ defined by the formula $f(x)=F(x)_{0}$ is the time-zero coder associated with F. Notice that stationary coding preserves ergodicity.

Let T be an invertible, ergodic transformation of the probability space $(X,\Sigma ,\mu )$ . Let S be a set of positive measure. Denote by $\{ R_{n}\} $ the $(T,\mathcal {A}_{S})$ -process defined by the partition $\mathcal {A}_{S}=\{ S,X\setminus S\} $ with $X\setminus S$ labeled by $0$ and S labeled by $1$ . That is, $R_{n}(x)=\chi _{S}(T^{n}x)$ for every integer $n\in \mathbb {Z}$ . Notice that $\{ R_{n}\} $ is a stationary coding of the $(T,\mathcal {A}_{S})$ -process with time zero coder $\chi _{S}$ and therefore it is ergodic.

Notice that such a non-trivial $\{ R_{n}\} $ process (i.e $0<\mu (S)<1$ ) has positive entropy where $(X,\Sigma ,\mu )$ is a product measure on $\mathcal {A}_{S}^{\mathbb {Z}}$ . This is because a non-trivial $(T,\{ S,X\setminus S\} )$ -process is a factor of Bernoulli measure which has completely positive entropy.

6.2 Reverse Markov’s inequality

The next reverse version of Markov’s inequality is famous and easy to prove. We present it here with a proof for completeness.

Theorem 6.1. (Reverse Markov’s inequality)

Let X be a random variable on a probability space $(\Omega ,\textbf {P})$ that satisfies $\textbf {P}(X\leq a)=1$ for some constant a. Then, for $d<\textbf {E}(X)$ ,

$$ \begin{align*} \textbf{P} (X>d )\geq\frac{\textbf{E}(X)-d}{a-d}. \end{align*} $$

Proof. Define the random variable $V=a-X$ which is almost surely non-negative by assumption. The event $\{ X\leq d\} $ is equivalent to the event $\{ V\geq a-d\} $ . Now with Markov’s inequality,

$$ \begin{align*} \textbf{P} ( \{ X\leq d \} )=\textbf{P} ( \{ V\geq a-d \} )\leq\frac{\textbf{E}(V)}{a-d}=\frac{a-\textbf{E}(X)}{a-d}, \end{align*} $$

where the right-hand side numerator and denominator are strictly positive since $d<\textbf {E}(X)\leq a$ . Finally,

$$ \begin{align*} \textbf{P} ( \{ X>d \} )=1-\textbf{P} ( \{ X\leq d \} )>\frac{\textbf{E}(X)-d}{a-d}.\\[-41pt] \end{align*} $$

6.3 Not all the $\mu $ values meet the condition in equation (4.2)

Here we construct a counterexample which violates the weaker condition in equation (4.2). We anticipate that the condition in equation (4.1) can be violated by simpler examples.

We use the formalism from §§2.2 and 6.1. Let $(X_{l},\Sigma ,\nu ,\sigma )$ be a uniformly distributed full Bernoulli shift in $2<l$ symbols. We define a measurable set $\mathcal {Y}\in \Sigma $ recursively.

Fix $0<\epsilon <0.5$ . For $k=1$ , fix $2\leq n_{1}\in \mathbb {N}$ such that $\log (n_{1})^{-1}<1-\epsilon $ and let $Y_{1}$ be a set such that $0.5\log (n_{1})^{-2}<\nu (Y_{1})<\log (n_{1})^{-2}$ . Define $\mathcal {Y}_{1}=\bigcup _{m=0}^{[\log n_{1}]}\sigma ^{-m}Y_{1}$ so that $0<\nu (\mathcal {Y}_{1})\leq \log (n_{1})^{-1}$ and also $\nu (\mathcal {Y}_{1})<1-\epsilon $ .

For $1<k\in \mathbb {N}$ , choose $n_{k-1}<n_{k}\in \mathbb {N}$ such that $\log (n_{k})^{-1}<1-\epsilon -\sum _{i=1}^{k-1}\nu (\mathcal {Y}_{i})$ and let $Y_{k}$ be a set such that $0.5\cdot \log (n_{k})^{-2}<\nu (Y_{k})<\log (n_{k})^{-2}$ . Define $\mathcal {Y}_{k}=\bigcup _{m=0}^{[\log n_{k}]}\sigma ^{-m}Y_{k}$ so that $0<\nu (\mathcal {Y}_{k})\leq {1}/{\log n_{k}}\leq 1-\epsilon -\sum _{i=1}^{k-1}\nu (\mathcal {Y}_{i})$ and therefore $0<\sum _{i=1}^{k}\nu (\mathcal {Y}_{i})<1-\epsilon $ .

Finally, define $\mathcal {Y}=\bigcup _{i=1}^{\infty }\mathcal {Y}_{i}$ and define a process on $\{ 0,1\} ^{\mathbb {Z}}$ by $R_{n}(x)=\chi _{\mathcal {Y}}(\sigma ^{n-1}x)$ with the induced measure $\widetilde {\mu }=\nu \circ R^{-1}$ as in §6.1. By our construction, $0<\nu (\mathcal {Y})<1-\epsilon $ and $\{ n_{k}\} $ is strictly increasing. Since $(X_{l},\Sigma ,\nu )$ is ergodic, so is $\{ R_{n}\} $ as a stationary coding with time zero coder (which trivially also preserves the measure). That is, we can interpret $\widetilde {\mu }$ as a probability measure which is ergodic and shift invariant on the binary full shift. It also has positive entropy by §6.1.

In another direction, notice that $[0,1)$ is the image of the infinite binary sequences $\{ 0,1\} ^{\mathbb {N}}$ under the natural map $(x_{i})_{i=1}^{\infty }\mapsto \sum _{i=1}^{\infty }({x_{i}}/{2^{i}})$ which is also a bijection on the complement of a countable set, that is, a bijection on the complement of a null set for every continuous measure. Thus, if we denote $X_{2}=\{ 0,1\} ^{\mathbb {Z}}$ (the 2-shift), then we can define up to a null set the bijection $\iota :X_{2}\rightarrow \{ 0,1\} ^{-\mathbb {N}_{0}}\times [0,1)$ . We can also define the pushforward $\widetilde {\rho }=\widetilde {\mu }\circ \iota ^{-1}$ (that is, a measure on $\{ 0,1\} ^{-\mathbb {N}_{0}}\times [0,1)$ ) which preserves all the relevant properties of $\widetilde {\mu }$ (ergodicity is trivial and positive entropy by being a non-trivial factor of the system with completely positive entropy). Now, $\widetilde {\rho }=\intop \!\rho _{\eta }\,d\widetilde {\rho }(\eta )$ , where it is disintegrated with respect to the $\sigma $ -algebra that is generated by projection to the past (like in §1.3) and the conditional measures $\rho _{\eta }$ identified as measures on $[0,1)$ . Similarly, $\widetilde {\mu }=\intop \! d\mu {}_{\eta }\,d\widetilde {\mu }(\eta )$ , where $\eta \in \{ 0,1\} ^{-\mathbb {N}_{0}}$ .

For every $k\in \mathbb {N}$ , denote the event $\bigcap _{i=0}^{[\log n_{k}]}\{ R_{i}=1\}$ by $E_{k}$ . We begin with

$$ \begin{align*} {\intop\!\iint\chi_{B_{n_{k}^{-1}}(y)}(x)\,d\rho_{\eta}(x)\,d\rho_{\eta}(y)\,d\widetilde{\rho}(\eta)}, \end{align*} $$

where $B_{n_{k}^{-1}}(y)=\{ x\in [0,1): |x-y |<n_{k}^{-1}\} $ . We need to show that this integral decays in $n_{k}$ with sub polynomial rate. We can pullback this integration to the space $X_{2}$ and restrict it to the event $E_{k}\times E_{k}$ since every pair $(x',y')\in E_{k}\times E_{k}$ corresponds to a pair $(x,y)\in [0,1)^{2}$ with $ |x-y |<{1}/{n_{k}}$ . Thus,

$$ \begin{align*}& \geq\intop\!\intop\!\!\!\intop\chi_{E_{k}\times E_{k}} (x,y )\,d\mu_{\eta}\times\mu_{\eta} (x,y )\,d\widetilde{\mu} (\eta )\\& =\intop\!\intop\!\chi_{E_{k}} (x )\,d\mu{}_{\eta} (x )\intop\!\chi_{E_{k}} (y )\,d\mu_{\eta} (y )\,d\widetilde{\mu} (\eta )\\& =\intop\!\! \bigg(\!\intop\!\chi_{E_{k}} (x )\,d\mu{}_{\eta} (x ) \bigg)^{2}\,d\widetilde{\mu} (\eta )\\& \geq \bigg(\!\intop\!\intop\!\chi_{E_{k}} (x )\,d\mu{}_{\eta} (x )\,d\widetilde{\mu} (\eta ) \bigg)^{2}\\& = \bigg(\!\intop\!\chi_{E_{k}} (x )\,d\widetilde{\mu} (x ) \bigg)^{2},\end{align*} $$

where in the first equality, we used Fubini and then split the integral into two separate integrals, and the second inequality is Jensen. We can pullback the integration once again to the full shift $X_{l}$ and restrict it to the event $Y_{k}$ which included in the preimage of $E_{k}$ ,

$$ \begin{align*} & \geq \bigg(\!\intop\!\chi_{Y_{k}} (x)\,d\nu (x ) \bigg)^{2}\\ & =0.25\cdot\log(n_{k})^{-4}. \end{align*} $$

Now, for every $k\in \mathbb {N}$ , we can use the reverse Markov inequality (Theorem 6.1) with $a=2$ and $d={1}/{8\log (n_{k})^{4}}$ to get

$$ \begin{align*} \textbf{P}_{\widetilde{\mu}} \bigg(\bigg\{\intop\!\mu{}_{\eta} (B_{n_{k}^{-1}} (y ) )\,d\mu{}_{\eta} (y )\geq\frac{1}{8\log (n_{k} )^{4}}\bigg\} \bigg)\geq\frac{1}{16\log (n_{k} )^{4}}. \end{align*} $$

This violates the condition in equation (4.2).

6.4 Processes with memory of finite length meet the condition in equation (4.1)

In this special case, it will be enough to use the total probability formula to reach the condition in equation (4.1) which also implies that the condition in equation (4.2) holds.

Let A be a finite alphabet $ |A |=l$ for some integer $l\geq 2$ and let $(A^{\mathbb {Z}},\mu ,T)$ be a finite memory length symbolic process which is ergodic, invariant, and with positive entropy with respect to T. That is, the prediction of $A_{0}$ depends only on a finite portion of the past $A_{-n},\ldots ,A_{-1}$ for some $n<\infty $ or explicitly for every $m\in \mathbb {Z}$ and $x\in A^{\mathbb {Z}}$ , $\mu ([x_{m+n}^{\infty }]|[x_{-\infty }^{m}])=\mu ([x_{m+n}^{\infty }])$ . Denote $s=\max _{a\in A}\mu ([a])<1$ . Assume that $n<m$ , then with direct computation,

$$ \begin{align*} \mu_{ [x_{-\infty}^{0} ]} ( [x_{0}^{m} ] ) & \leq\mu ( [x_{n}^{m} ] )\\ & =\mu ( [x_{n} ] )\mu ( [x_{n+1}^{m} ]| [x_{n} ] )\\& \leq\mu ( [x_{n} ] )\mu ( [x_{2n}^{m} ] )\\ & \vdots\\ & =\prod_{i=1}^{[m/n]}\mu ( [x_{n\cdot i} ] )\\ & \leq s^{m/n+1}. \end{align*} $$

If we consider a uniform partition of the unit interval $\{ I_{j}^{k}\} _{j=1}^{k}$ such that $ |I_{j}^{k} |={1}/{k}$ and where $k=l^{m}$ , then the display above implies that $\mathop {}\!\mathrm {ess\,sup}_{j,\omega }\mu _{\omega }(I_{j}^{k})\leq O(k^{\log (s)/(n\log (l))})$ .

7.1 Self-similarity and classes of algebraic numbers

Here we set out some of the basic definitions regarding self similarity. A set $\mathcal {F}=\{ f_{1},\ldots f_{k}\} $ for $k\geq 2$ of contractions on ${\mathbb {R}}$ , $f_{i}(x)=r_{i}x+a_{i}$ , with $0<r_{i}<1$ for each $i\in \{ 1,\ldots ,k\} $ , is an iterated function system or IFS for short. We also call the $f_{i}$ terms similarities.

A key fact in this topic is that there exists a unique non-empty compact set $K\subset {\mathbb {R}}$ such that $K=\bigcup _{i=1}^{k}f_{i}(K)$ . We call it the attractor or the self-similar set of the IFS $\mathcal {F}$ .

Given a list of positive numbers $\boldsymbol {p}=(p_{1},\ldots ,p_{k})$ with $\sum _{i=1}^{k}p_{i}=1$ , we call it a positive probability vector and there is a unique probability measure $\mu _{\boldsymbol {p}}$ with $\mu _{\boldsymbol {p}}=\sum _{i=1}^{k}p_{i}\cdot f_{i}\mu _{\boldsymbol {p}}$ . This measure is supported on the attractor of $\mathcal {F}$ and called a self-similar measure.

A Pisot number is a real algebraic integer greater than $1$ all of whose Galois conjugates are less than $1$ in absolute value. A Salem number is a real algebraic integer greater than $1$ whose all conjugate roots have absolute value no greater than $1$ , and at least one of them has an absolute value which equals $1$ . There are countably many Pisot and Salem numbers.

We conclude this part with a recent theorem of Varjú and Yu [Reference Varjú and Yu13] which plays a major role in our proof that $T_{b}$ -invariant Rajchman measure need not be Parry.

Theorem 7.1. Let $k\geq 2$ be an integer. Let $r_{1}=r^{l_{1}},\ldots ,r_{k}=r^{l_{k}}$ for some $r\in (0,1)$ and $l_{1},\ldots ,l_{k}\in \mathbb {Z}_{>0}$ with $\mathop{\mathrm{gcd}} (l_{1},\ldots ,l_{k})=1$ . Assume that $r^{-1}$ is not a Pisot or Salem number. Let $\mu $ be a non-singleton self-similar measure associated to the IFS $\mathcal {F}=\{ f_{1},\ldots ,f_{k}\} $ ( $f_{i}(x)=r_{i}x+a_{i}$ ) and a positive probability measure. Then,

$$ \begin{align*} |\hat{\mu} (z ) |=O ( |\log (z )^{-c} | ) \end{align*} $$

for some $c>0.$

We use Theorem 7.1 to prove that a Rajchman $T_{b}$ -invariant measure must not be unique. Let $b\geq 2$ be a specified non-simple number which is not Salem or Pisot. We can guarantee the existence of such a number by cardinality considerations (recall that Pisot numbers as well as Salem numbers and simple numbers are countable while specified numbers have the cardinality of continuum).

Denote $\mathbb {N}_{0}=\mathbb {N}\cup \{ 0\} $ for short and denote the b-expansion of b by $(a_{i})$ . That is, $b=a_{0}+{a_{1}}/{b}+\cdots $ and in particular $a_{0}=[b]$ . Every sequence $(x_{i})\in \{ 0,1\} ^{\mathbb {N}_{0}}$ and $n\in \mathbb {N}$ satisfies $(x_{i+n})<(a_{i})$ since $x_{n}\in \{ 0,1\} $ which is either way less than $[b]=a_{0}$ . Recall that $X_{b}\subset \Lambda _{b}^{\mathbb {N}_{0}}$ is the subset of sequences that encodes b-expansions, so by Parry’s criterion (Theorem 2.2), we get that $\{ 0,1\} ^{\mathbb {N}_{0}}\subset X_{b}$ .

Define the set $K=\{ \sum _{i=0}^{\infty }({b_{i}}/{b^{i}}):(b_{i})\in \{ 0,1\} ^{\mathbb {N}_{0}}\} $ which is the image of $\{ 0,1\} ^{\mathbb {N}}$ under the b-expansion and define the IFS $\mathcal {F}=\{ f_{0}(x)={x}/{b},f_{1}(x)={x}/{b}+{1}/{b}\} $ . If we denote concatenation of symbols by $\centerdot $ , then clearly $\{ 0,1\} ^{\mathbb {N}_{0}}=0\centerdot \{ 0,1\} ^{\mathbb {N}_{0}}\cup 1\centerdot \{ 0,1\} ^{\mathbb {N}_{0}}$ , which means under the b-expansion, $K=\bigcup _{i=0}^{1}f_{i}(K)$ . Thus, K is the attractor of $\mathcal {F}$ and for every positive probability vector $\boldsymbol {p}=(p_{0},p_{1})$ , the self-similar measure that satisfies $\mu _{\boldsymbol {p}}=\sum _{i=0}^{1}p_{i}\cdot f_{i}\mu _{\boldsymbol {p}}$ is well defined. Pulling it back again under the b-expansion, it translated into a Bernoulli shift $\boldsymbol {p}^{\mathbb {N}_{0}}$ on $X_{b}|_{\{ 0,1\} ^{\mathbb {N}_{0}}}=\{ 0,1\} ^{\mathbb {N}_{0}}$ , where the former is $T_{b}$ -invariant and correspondingly the latter is shift invariant.

If we consider Theorem 7.1 where both of the $l_{i}$ terms are equal to $1$ such that $\mathrm{gcd}(l_{0},l_{1})=1$ , then all the hypotheses hold with respect to b and $\mathcal {F}$ and any self-similar measure $\mu _{\boldsymbol {p}}$ provides the Rajchman property,

$$ \begin{align*} |\hat{\mu}_{\boldsymbol{p}} (z ) |=O ( |\log (z )^{-c} | ). \end{align*} $$

Since there are infinitely many of them, the uniqueness is violated.