Expansions of real numbers with respect to two integer bases
Jörg Neunhäuserer
本文研究了实数对两个整数基的展开方法及其性质。
我们介绍并研究了实数对两个整数基的展开。
We introduce and study expansions of real numbers with respect to two integer bases.
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In this paper we study the set of digit frequencies that are realised by elements of the set of $β$-expansions. The main result of this paper demonstrates that as $β$ approaches $1,$ the set of digit frequencies that occur amongst the set of $β$-expansions fills out the simplex. As an application of our main result, we obtain upper bounds for the local dimension of certain biased Bernoulli convolutions.
For a given positive integer $m$, let $A=\set{0,1,...,m}$ and $q \in (m,m+1)$. A sequence $(c_i)=c_1c_2 ...$ consisting of elements in $A$ is called an expansion of $x$ if $\sum_{i=1}^{\infty} c_i q^{-i}=x$. It is known that almost every $x$ belonging to the interval $[0,m/(q-1)]$ has uncountably many expansions. In this paper we study the existence of expansions $(d_i)$ of $x$ satisfying the inequalities $\sum_{i=1}^n d_iq^{-i} \geq \sum_{i=1}^n c_i q^{-i}$, $n=1,2,...$ for each expansion $(c_i)$ of $x$.
Let $q > 1$ be a real number and let $m=m(q)$ be the largest integer smaller than $q$. It is well known that each number $x \in J_q:=[0, \sum_{i=1}^{\infty} m q^{-i}]$ can be written as $x=\sum_{i=1}^{\infty}{c_i}q^{-i}$ with integer coefficients $0 \le c_i < q$. If $q$ is a non-integer, then almost every $x \in J_q$ has continuum many expansions of this form. In this note we consider some properties of the set $\mathcal{U}_q$ consisting of numbers $x \in J_q$ having a unique representation of this form. More specifically, we compare the size of the sets $\mathcal{U}_q$ and $\mathcal{U}_r$ for…
Let $q\in(1,2)$ and $x\in[0,\frac1{q-1}]$. We say that a sequence $(\varepsilon_i)_{i=1}^{\infty}\in\{0,1\}^{\mathbb{N}}$ is an expansion of $x$ in base $q$ (or a $q$-expansion) if \[ x=\sum_{i=1}^{\infty}\varepsilon_iq^{-i}. \] For any $k\in\mathbb N$, let $\mathcal B_k$ denote the set of $q$ such that there exists $x$ with exactly $k$ expansions in base $q$. In [12] it was shown that $\min\mathcal B_2=q_2\approx 1.71064$, the appropriate root of $x^{4}=2x^{2}+x+1$. In this paper we show that for any $k\geq 3$, $\min\mathcal B_k=q_f\approx1.75488$, the appropriate root of $x^3=2x^2-x+1$.
We introduce and study non-uniform expansions of real numbers, given by two non-integer bases.
We obtain estimates for the number of integral solutions in large balls, of inequalities of the form $|Q(x, y)| < ε$, where $Q$ is an indefinite binary quadratic form, in terms of the Hurwitz continued fraction expansions of the slopes of the lines on which $Q$ vanishes. The method is based on a coding of geodesics on the modular surface via Hurwitz expansions of the endpoints of their lifts in the Poincare half-plane.
Paul Erdos asked how frequently the ternary expansion of 2^n omits the digit 2. He conjectured this happens only for finitely many values of n. We generalize this question to consider iterates of two discrete dynamical systems. The first is over the real numbers, and considers the integer part of lambda 2^n for a real input lambda. The second is over the 3-adic integers, and considers the sequence lambda 2^n for a 3-adic integer input lambda. We show that the number of input values that have infinitely many iterates omitting the digit 2 in their ternary expansion is small in a suitable sense. …
Let $b\geq 2$ be an integer and $\hv$ a real number. Among other results, we compute the Hausdorff dimension of the set of real numbers $ξ$ with the property that, for every sufficiently large integer $N$, there exists an integer $n$ such that $1 \le n \le N$ and the distance between $b^n ξ$ and its nearest integer is at most equal to $b^{-\hv N}$. We further solve the same question when replacing $b^nξ$ by $T^n_βξ$, where $T_β$ denotes the classical $β$-transformation.
A folklore conjecture in number theory states that the only integers whose expansions in base $3,4$ and $5$ contain solely binary digits are $0, 1$ and $82000$. In this paper, we present the first progress on this conjecture. Furthermore, we investigate the density of the integers containing only binary digits in their base $3$ or $4$ expansion, whereon an exciting transition in behaviour is observed. Our methods shed light on the reasons for this, and relate to several well-known questions, such as Graham's problem and a related conjecture of Pomerance. Finally, we generalise this setting and…
Given a positive integer $M$ and $q\in(1,M+1]$, let $\mathcal U_q$ be the set of $x\in[0, M/(q-1)]$ having a unique $q$-expansion: there exists a unique sequence $(x_i)=x_1x_2\ldots$ with each $x_i\in\{0,1,\ldots, M\}$ such that \[ x=\frac{x_1}{q}+\frac{x_2}{q^2}+\frac{x_3}{q^3}+\cdots. \] Denote by $\mathbf U_q$ the set of corresponding sequences of all points in $\mathcal U_q$. It is well-known that the function $H: q\mapsto h(\mathbf U_q)$ is a Devil's staircase, where $h(\mathbf U_q)$ denotes the topological entropy of $\mathbf U_q$. In this paper we {give several characterizations of} the…
In this paper, we consider some additive properties of integers with restricted digit expansions. Let $b\geq 3$ be an integer and $B_b$ be the set of integers whose base $b$ expansions have only digits $\{0,1\}.$ Let $a,b,c$ be three integers greater than $2.$ We give some estimates on the size of $(B_{a}+B_{b})\cap B_{c}.$ In particular, under mild conditions, $(B_{a}+B_{b})\cap B_{c}$ is a very thin set in the following sense that for each $ε>0,$ as $N\to\infty,$ \[ \#((B_{a}+B_{b})\cap B_{c} \cap [1,N])=O(N^ε). \]
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…
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.
For a positive integer $M$ and a real base $q\in(1,M+1]$, let $\mathcal{U}_q$ denote the set of numbers having a unique expansion in base $q$ over the alphabet $\{0,1,\dots,M\}$, and let $\mathbf{U}_q$ denote the corresponding set of sequences in $\{0,1,\dots,M\}^{\mathbb{N}}$. Komornik et al. [Adv. Math. 305 (2017), 165--196] showed recently that the Hausdorff dimension of $\mathcal{U}_q$ is given by $h(\mathbf{U}_q)/\log q$, where $h(\mathbf{U}_q)$ denotes the topological entropy of $\mathbf{U}_q$. They furthermore showed that the function $H: q\mapsto h(\mathbf{U}_q)$ is continuous, nondecr…
We show that for a Salem number $β$ of degree $d$, there exists a positive constant $c(d)$ that $β^m$ is a Parry number for integers $m$ of natural density $\ge c(d)$. Further, we show $c(6)>1/2$ and discuss a relation to the discretized rotation in dimension $4$.
In the present article, we introduce beta-expansions in the ring $\mathbb{Z}_p$ of $p$-adic integers. We characterise the sets of numbers with eventually periodic and finite expansions.
Spectral and dynamical results related to certain non-integer base expansions on the unit interval
本文研究了非整数底β展开相关的转移算子及其伴随算子的性质，证明了在Lipschitz条件下，迭代序列在L1范数下以指数速度收敛到与算子的特征值1对应的不变态。
We consider digits-deleted sets or Cantor-type sets with $β$-expansions. We calculate the Hausdorff dimension $d$ of these sets and show that $d$ is continuous with respect to $β$. The $d$-dimentional Hausdorff measure of these sets is finite and positive.
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