Let $β>1$ and let $m>\be$ be an integer. Each $x\in I_\be:=[0,\frac{m-1}{β-1}]$ can be represented in the form \[ x=\sum_{k=1}^\infty ε_kβ^{-k}, \] where $ε_k\in\{0,1,...,m-1\}$ for all $k$ (a $β$-expansion of $x$). It is known that a.e. $x\in I_β$ has a continuum of distinct $β$-expansions. In this paper we prove that if $β$ is a Pisot number, then for a.e. $x$ this continuum has one and the same growth rate. We also link this rate to the Lebesgue-generic local dimension for the Bernoulli convolution parametrized by $β$.
When $β<\frac{1+\sqrt5}2$, we show that the set of $β$-expansions grows exponentially for every internal $x$.
This paper studies tilings related to the beta-transformation when beta is a Pisot number (that is not supposed to be a unit). Then it applies the obtained results to study the set of rational numbers having a purely periodic beta-expansion. Special focus is given to some quadratic examples.
In a recent paper of Feng and Sidorov they show that for $β\in(1,\frac{1+\sqrt{5}}{2})$ the set of $β$-expansions grows exponentially for every $x\in(0,\frac{1}{β-1})$. In this paper we study this growth rate further. We also consider the set of $β$-expansions from a dimension theory perspective.
The structures of full words and non-full for $β$-expansions are completely characterized in this paper. We obtain the precise lengths of all the maximal runs of full and non-full words among admissible words with same order.
We study the typical growth rate of the number of words of length n which can be extended to beta-expansions of x. In the general case we give a lower bound for the growth rate, while in the case that the Bernoulli convolution associated to parameter beta is absolutely continuous we are able to give the growth rate precisely. This gives new necessary and sufficient conditions for the absolute continuity of Bernoulli convolutions.
The finiteness property is an important arithmetical property of beta-expansions. We exhibit classes of Pisot numbers $β$ having the negative finiteness property, that is the set of finite $(-β)$-expansions is equal to $\mathbb{Z}[β^{-1}]$. For a class of numbers including the Tribonacci number, we compute the maximal length of the fractional parts arising in the addition and subtraction of $(-β)$-integers. We also give conditions excluding the negative finiteness property.
Let β_n>1 be a root of x^n-x-1 for n=4,5,... We will prove that β_n is not a Parry number, i.e., the associated beta transformation does not correspond a sofic symbolic system. A generalization is shown in the last section.
Let $1<β\leq 2$. It is well-known that the set of points in $% [0,1/(β-1)]$ having unique $β$-expansion, in other words, those points whose orbits under greedy $β$-transformation escape a hole depending on $β$, is of zero Lebesgue measure. The corresponding escape rate is investigated in this paper. A formula which links the Hausdorff dimension of univoque set and escape rate is established in this study. Then we also proved that such rate forms a devil's staircase function with respect to $β$.
We study natural measures on sets of beta-expansions and on slices through self similar sets. In the setting of beta-expansions, these allow us to better understand the measure of maximal entropy for the random beta-transformation and to reinterpret a result of Lindenstrauss, Peres and Schlag in terms of equidistribution. Each of these applications is relevant to the study of Bernoulli convolutions. In the fractal setting this allows us to understand how to disintegrate Hausdorff measure by slicing, leading to conditions under which almost every slice through a self similar set has positive Ha…
Let $β>1$ be a non-integer. First we show that Lebesgue almost every number has a $β$-expansion of a given frequency if and only if Lebesgue almost every number has infinitely many $β$-expansions of the same given frequency. Then we deduce that Lebesgue almost every number has infinitely many balanced $β$-expansions, where an infinite sequence on the finite alphabet $\{0,1,\cdots,m\}$ is called balanced if the frequency of the digit $k$ is equal to the frequency of the digit $m-k$ for all $k\in\{0,1,\cdots,m\}$. Finally we consider variable frequency and prove that for every pseudo-golden rati…
Let $β> 1$ and the run-length function $r_n(x,β)$ be the maximal length of consecutive zeros amongst the first n digits in the $β$-expansion of $x\in[0,1]$. The exceptional set $$E_{\max}^φ=\left\{x \in [0,1]:\liminf_{n\rightarrow \infty}\frac{r_n(x,β)}{φ(n)}=0, \limsup_{n\rightarrow \infty}\frac{r_n(x,β)}{φ(n)}=+\infty\right\}$$ is investigated, where $φ: \mathbb{N} \rightarrow \mathbb{R}^+$ is a monotonically increasing function with $\lim\limits_{n\rightarrow \infty }φ(n)=+\infty$. We prove that the set $E_{\max}^φ$ is either empty or of full Hausdorff dimension and residual in $[0,1]$ acco…
Let $\be\in(1,2)$. Each $x\in I_\be:=[0,\frac{1}{\be-1}]$ can be represented in the form \[ x=\sum_{k=1}^\infty a_k\be^{-k}, \] where $a_k\in\{0,1\}$ for all $k$ (a $\be$-expansion of $x$). It was shown in \cite{S} that a.e. $x\in I_\be$ has a continuum of distinct $\be$-expansions. In this paper we show that for a generic $x$, this continuum has one and the same growth rate, i.e., the general $\be$-expansions exhibit an ergodic behaviour. When $\be<\frac{1+\sqrt5}2$, we show that the set of $\be$-expansions grows exponentially for every $x\in(0,\frac{1}{\be-1})$. Special attention is paid to …
Let $β\in(1,2)$ be a Pisot unit and consider the symmetric $β$-expansions. We give a necessary and sufficient condition for the associated Rauzy fractals to form a tiling of the contractive hyperplane. For $β$ a $d$-Bonacci number, i.e., Pisot root of $x^d-x^{d-1}-\dots-x-1$ we show that the Rauzy fractals form a multiple tiling with covering degree $d-1$.
Let $β>1$, $I$ be the unite interval $[0,1)$ and $ϕ$ be an integer function defined on $\mathbb{N}\setminus\{0\}$ satisfying $1\leqϕ(n)\leq n$. Denote by $A_ϕ(x,β)$ the Erdös-Rényi average of $x\in I$ associated with the function $ϕ$ in $β$-expansion and $I_β$ the range of $A_ϕ(x,β)$ for $x\in I$. For the level set \begin{align*} ER_ϕ^β(α)=\left\{x\in I\colon A_ϕ(x,β)=α\right\},\quad\text{where}\ α\in I_β, \end{align*} in this paper we will determine its Hausdorff dimension under the assumption $ϕ(n)\to\infty$ as $n\to\infty$ and $ϕ$ is the integer part of some slowly varying sequence. Besides…
By applying a 2014 result on the distribution of full cylinders, we give a proof of the useful folklore: for any $β>1$, the Hausdorff dimension of an arbitrary set in the shift space $S_β$ is equal to the Hausdorff dimension of its natural projection in $[0,1]$. It has been used in some former papers without proof. Then we clarify that for calculating the Hausdorff dimension of frequency sets using variational formulae, one only needs to focus on the Markov measures of explicit order when the $β$-expansion of $1$ is finite. Concretely, it suffices to optimize a function with finitely many vari…
Let $β\in(1,2)$. Each $x\in[0,\frac{1}{β-1}]$ can be represented in the form \[ x=\sum_{k=1}^\infty ε_kβ^{-k}, \] where $ε_k\in\{0,1\}$ for all $k$ (a $β$-expansion of $x$). If $β>\frac{1+\sqrt5}{2}$, then, as is well known, there always exist $x\in(0,\frac1{β-1})$ which have a unique $\be$-expansion. In the present paper we study (purely) periodic unique $β$-expansions and show that for each $n\ge2$ there exists $β_n\in[\frac{1+\sqrt5}{2},2)$ such that there are no unique periodic $β$-expansions of smallest period $n$ for $β\leβ_n$ and at least one such expansion for $β>β_n$. Furthermore, we …
We study a family of piecewise expanding maps on the plane, generated by composition of a rotation and an expansive similitude of expansion constant $β$. We give two constants $B_1$ and $B_2$ depending only on the fundamental domain that if $β>B_1$ then the expanding map has a unique absolutely continuous invariant probability measure, and if $β>B_2$ then it is equivalent to $2$-dimensional Lebesgue measure. Restricting to a rotation generated by $q$-th root of unity $ζ$ with all parameters in $\mathbb{Q}(ζ,β)$, it gives a sofic system when $\cos(2π/q) \in \mathbb{Q}(β)$ and $β$ is a Pisot num…
From the works of Rauzy and Thurston, we know how to construct (multiple) tilings of some Euclidean space using the conjugates of a Pisot unit $β$ and the greedy $β$-transformation. In this paper, we consider different transformations generating expansions in base $β$, including cases where the associated subshift is not sofic. Under certain mild conditions, we show that they give multiple tilings. We also give a necessary and sufficient condition for the tiling property, generalizing the weak finiteness property (W) for greedy $β$-expansions. Remarkably, the symmetric $β$-transformation does …
We study rational numbers with purely periodic Rényi $β$-expansions. For bases $β$ satisfying $β^2=aβ+b$ with $b$ dividing $a$, we give a necessary and sufficient condition for $γ(β)=1$, i.e., that all rational numbers $p/q\in[0,1)$ with $\gcd(q,b)=1$ have a purely periodic $β$-expansion. A simple algorithm for determining the value of $γ(β)$ for all quadratic Pisot numbers $β$ is described.
In the base phi representation any natural number is written uniquely as a sum powers of the golden mean with digits 0 and 1, where one requires that the product of two consecutive digits is always 0. In this paper we give precise expressions for the those natural numbers for which the $k$th digit is 1, proving two conjectures for $k=0,1$. The expressions are all in terms of generalized Beatty sequences.
The beta-conjugates of a base of numeration $β> 1$, $β$ being a Parry number, were introduced by Boyd, in the context of the Rényi-Parry dynamics of numeration system and the beta-transformation. These beta-conjugates are canonically associated with $β$. Let $β> 1$ be a real algebraic number. A more general definition of the beta-conjugates of $β$ is introduced in terms of the Parry Upper function $f_β(z)$ of the beta-transformation. We introduce the concept of a germ of curve at $(0,1/β) \in \mathbb{C}^{2}$ associated with $f_β(z)$ and the reciprocal of the minimal polynomial of $β$. This ger…
We study invariant measures of a piecewise expanding map in $\mathbb{R}^m$ defined by an expanding similitude modulo lattice. Using the result of Bang on a problem of Tarski, we show that when the similarity ratio is not less than $m+1$, it has an absolutely continuous invariant measure equivalent to the $m$-dimensional Lebesgue measure, under some mild assumption on the fundamental domain. Applying the method to the case $m=2$, we obtain an alternative proof of the result in Akiyama-Caalim:2015 together with some improvement.
For $β> 1$, a sequence $(c_n)_{n \geq 1} \in \mathbb{Z}^{\mathbb{N}^+}$ with $0 \leq c_n < β$ is the \emph{beta expansion} of $x$ with respect to $β$ if $x = \sum_{n = 1}^\infty c_nβ^{-n}$. Defining $d_β(x)$ to be the greedy beta expansion of $x$ with respect to $β$, it is known that $d_β(1)$ is eventually periodic as long as $β$ is a Pisot number. It is conjectured that the same is true for Salem numbers, but is only currently known to be true for Salem numbers of degree 4. Heuristic arguments suggest that almost all degree 6 Salem numbers admit periodic expansions but that a positive proport…
Let $β> 1$ be a real number and $x \in [0,1)$ be an irrational number. Denote by $k_n(x)$ the exact number of partial quotients in the continued fraction expansion of $x$ given by the first $n$ digits in the $β$-expansion of $x$ ($n \in \mathbb{N}$). In this paper, we show a central limit theorem and a law of the iterated logarithm for the random variables sequence $\{k_n, n \geq 1\}$, which generalize the results of Faivre and Wu respectively from $β=10$ to any $β>1$.
Let $β> 1$ be a real number and $x \in [0,1)$ be an irrational number. We denote by $k_n(x)$ the exact number of partial quotients in the continued fraction expansion of $x$ given by the first $n$ digits in the $β$-expansion of $x$ ($n \in \mathbb{N}$). It is known that $k_n(x)/n$ converges to $(6\log2\logβ)/π^2$ almost everywhere in the sense of Lebesgue measure. In this paper, we improve this result by proving that the Lebesgue measure of the set of $x \in [0,1)$ for which $k_n(x)/n$ deviates away from $(6\log2\logβ)/π^2$ decays to 0 exponentially as $n$ tends to $\infty$, which generalizes …
Shift radix systems form a collection of dynamical systems depending on a parameter $\mathbf{r}$ which varies in the $d$-dimensional real vector space. They generalize well-known numeration systems such as beta-expansions, expansions with respect to rational bases, and canonical number systems. Beta-numeration and canonical number systems are known to be intimately related to fractal shapes, such as the classical Rauzy fractal and the twin dragon. These fractals turned out to be important for studying properties of expansions in several settings. In the present paper we associate a collection …
Hausdorff Dimension of Growth Rate Level Sets in $θ$-expansions
National University of Science and Technology POLITEHNICA Bucharest（布加勒斯特理工大学）；Gheorghe Mihoc-Caius Iacob Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy（罗马尼亚科学院格奥尔基·米霍克-卡尤斯·雅各布数学统计与应用数学研究所）
The main purpose of this paper is to study the arithmetical properties of values \(\sum_{m=0}^{\infty} β^{-w(m)}\), where \(β\) is a fixed Pisot or Salem number and \(w(m)\) (\(m=0,1,\ldots\)) are distinct sequences of nonnegative integers with \(w(m+1)>w(m)\) for any sufficiently large \(m\). We first introduce criteria for the algebraic independence of such values. Our criteria are applicable to certain sequences \(w(m)\) (\(m=0,1,\ldots\)) with \(\lim_{m\to\infty}w(m+1)/w(m)=1.\) For example, we prove that two numbers \[\sum_{m=1}^{\infty}β^{-\lfloor φ(1,0;m)\rfloor}, \sum_{m=3}^{\infty}β^{…
We give explicit versions of Helfgott's Growth Theorem for $\SL_2$, as well as of the Bourgain-Gamburd argument for expansion of Cayley graphs modulo primes of subgroups of $\SL_2(\Zz)$ which are Zariski-dense in $\SL_2$.
