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We study the negative beta transformations $T_{-\beta}:=-\beta x +\lfloor\beta x\rfloor+1$ for $x\in(0,1]$ and $\beta>1$. We present a complete characterization of pairs of dstinct non-integers with the same $T_{-\beta}$-invariant measure:…
We analyse dynamical properties of the negative beta transformation, which has been studied recently by Ito and Sadahiro. Contrary to the classical beta transformation, the density of the absolutely continuous invariant measure of the…
Given $\beta\in(1,2)$, a $\beta$-expansion of a real $x$ is a power series in base $\beta$ with coefficients 0 and 1 whose sum equals $x$. The aim of this note is to study certain problems related to the universality and combinatorics of…
In this paper, we presented a new type of metric space called $(\alpha,\beta)$-metric space along with some novel contraction mappings named $(\alpha,\beta)$-contraction and weak $(\alpha,\beta)$-contraction mapping. We established some…
Among all positional numeration systems, the widely studied Bertrand numeration systems are defined by a simple criterion in terms of their numeration languages. In 1989, Bertrand-Mathis characterized them via representations in a real base…
We study the question of pure periodicity of expansions in the negative base numeration system. In analogy of Akiyama's result for positive Pisot unit base $\beta$, we find a sufficient condition so that there exist an interval $J$…
This paper studies the recursion-theoretic aspects of large-scale geometries of infinite strings, a subject initiated by Khoussainov and Takisaka (2017). We investigate several notions of quasi-isometric reductions between recursive…
We study $\alpha$-adic expansions of numbers in an extension field, that is to say, left infinite representations of numbers in the positional numeration system with the base $\alpha$, where $\alpha$ is an algebraic conjugate of a Pisot…
We present a study of the problem of finiteness of the $\beta$-expansions for the set of natural numbers, condition $F_1$ in brief, for three families of Pisot numbers for which the $\beta$-expansion of 1 is not a non-decreasing sequence.…
We introduce and study series expansions of real numbers with an arbitrary Cantor real base $\boldsymbol{\beta}=(\beta_n)_{n\in\mathbb{N}}$, which we call $\boldsymbol{\beta}$-representations. In doing so, we generalize both representations…
Good integers introduced in 1997 form an interesting family of integers that has been continuously studied due to their rich number theoretical properties and wide applications. In this paper, we have focused on classes of $2^\beta$-good…
It is known that for a uniform morphic sequence $\boldsymbol u = \langle u_n\rangle_{n=0}^\infty$ and an algebraic number $\beta$ such that $|\beta|>1$, the number $[\![\boldsymbol{u} ]\!]_\beta:=\sum_{n=0}^\infty \frac{u_n}{\beta^n}$…
For a given base $\gamma$ and a digit set ${\mathcal B}$ we consider optimal representations of a number $x$, as defined by Dajani at al. in 2012. For a non-integer negative base $\gamma=-\beta<-1$ and the digit set ${\mathcal…
Two infinite sequences A and B of non-negative integers are called additive complements, if their sum contains all sufficiently large integers. Let $A(x)$ and $B(x)$ be the counting functions of A and B. In this paper, we extend the results…
We show that for some constant $\beta > 0$, any subset $A$ of integers $\{1,\ldots,N\}$ of size at least $2^{-O((\log N)^\beta)} \cdot N$ contains a non-trivial three-term arithmetic progression. Previously, three-term arithmetic…
In this paper we study digit frequencies in the setting of expansions in non-integer bases, and self-affine sets with non-empty interior. Within expansions in non-integer bases we show that if $\beta\in(1,1.787\ldots)$ then every…
We prove that for any integers $\alpha, \beta > 1$, the existential fragment of the first-order theory of the structure $\langle \mathbb{Z}; 0,1,<, +, \alpha^{\mathbb{N}}, \beta^{\mathbb{N}}\rangle$ is decidable (where $\alpha^{\mathbb{N}}$…
This paper highlights an unexpected connection between expansions of real numbers to noninteger bases (so-called {\em $\beta$-expansions}) and the infinite derivatives of a class of self-affine functions. Precisely, we extend Okamoto's…
The zeta-dimension of a set A of positive integers is the infimum s such that the sum of the reciprocals of the s-th powers of the elements of A is finite. Zeta-dimension serves as a fractal dimension on the positive integers that extends…
In this paper we provide sufficient conditions in order to show that the set image of a continuous and shift-commuting map defined on a shift space over an arbitrary discrete alphabet is also a shift space; additionally, if such a map is…