Related papers: On digit frequencies in {\beta}-expansions

We determine the asymptotics of the number of independent sets of size $\lfloor \beta 2^{d-1} \rfloor$ in the discrete hypercube $Q_d = \{0,1\}^d$ for any fixed $\beta \in [0,1]$ as $d \to \infty$, extending a result of Galvin for $\beta…

The random beta polytope is defined as the convex hull of $n$ independent random points with the density proportional to $(1-\|x\|^2)^\beta$ on the $d$-dimensional unit ball, where $\beta>-1$ is a parameter. Similarly, the random beta'…

In this paper we classify the Zeckendorf expansions according to their digit blocks. It turns out that if we consider these digit blocks as labels on the Fibonacci tree, then the numbers ending with a given digit block in their Zeckendorf…

While faces of a polytope form a well structured lattice, in which faces of each possible dimension are present, this is not true for general compact convex sets. We address the question of what dimensional patterns are possible for the…

We study the relationship between the frequency of a ternary digit in a number and the asymptotic mean value of the digits. The conditions for the existence of the asymptotic mean of digits in a ternary number are established. We indicate…

Anomalous dynamics in which local perturbations spread faster than diffusion are ubiquitously observed in the long-time behavior of a wide variety of systems. Here, the manner by which such systems evolve towards their asymptotic…

Let $\eta=(\eta(t))_{t\in T}$ be a sample continuous max-infinitely random field on a locally compact metric space $T$. For a closed subset $S\in T$, we note $\eta_{S}$ the restriction of $\eta$ to $S$. We consider $\beta(S_1,S_2)$ the…

We introduce a novel deterministic fractal set PF in the unit interval whose construction is driven by the sequence of prime numbers modulo 16. At each step of the recursive construction, two subintervals are retained based on the residues…

We establish sharp bounds for the Hausdorff dimension of sets of irrational numbers in $(0,1)$ whose digits in the $N$-expansion are either uniformly bounded or tend to infinity. For sets with digits bounded by an integer $M \ge N$, we…

We study the convergence of certain subseries of the harmonic series corresponding to increasing sequences of integers whose digits in a certain base are not uniformly distributed. We also discuss the case of irregular sequences, where the…

We show the existence of Lebesgue-equivalent conservative and ergodic $\sigma$-finite invariant measures for a wide class of one-dimensional random maps consisting of piecewise convex maps. We also estimate the size of invariant measures…

In this paper, we introduce and investigate multivariate versions of frequent stability and diam-mean equicontinuity. Given a natural number $m > 1$, we call those notions "frequent $m$-stability" and "diam-mean $m$-equicontinuity". We use…

This summary of the doctoral thesis provides a comprehensive formulation of the Extended Discrete Fourier Transform (EDFT), derived directly from the Fourier integral and its orthogonality properties. The method is obtained by solving…

Properties of the set $T_s$ of "particularly non-normal numbers" of the unit interval are studied in details ($T_s$ consists of real numbers $x$, some of whose s-adic digits have the asymptotic frequencies in the nonterminating $s-$ adic…

Let $\beta >1$ be a non-integer. We consider expansions of the form $\sum_{i=1}^{\infty} d_i \beta^{-i}$, where the digits $(d_i)_{i \geq 1}$ are generated by means of a Borel map $K_{\beta}$ defined on $\{0,1\}^{\N}\times [ 0, \lfloor…

We consider the continued fraction digits as random variables measured with respect to Lebesgue measure. The logarithmically scaled and normalized fluctuation process of the digit sums converges strongly distributional to a random variable…

Let the base $\beta$ be a complex number, $|\beta|>1$, and let $A \subset \C$ be a finite alphabet of digits. The \emph{$A$-spectrum} of $\beta$ is the set $S_{A}(\beta) = \{\sum_{k=0}^n a_k\beta^k \mid n \in \mathbb{N}, \ a_k \in {A}\}$.…

Let $f(t_1,\ldots,t_n)$ be a nondegenerate integral quadratic form. We analyze the asymptotic behavior of the function $D_f(X)$, the number of integers of absolute value up to $X$ represented by $f$. When $f$ is isotropic or $n$ is at least…

Let $\beta > 1$ 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 $\beta$-expansion of $x$…

The beta polytope $P_{n,d}^\beta$ is the convex hull of $n$ i.i.d. random points distributed in the unit ball of $\mathbb{R}^d$ according to a density proportional to $(1-\lVert{x}\rVert^2)^{\beta}$ if $\beta>-1$ (in particular, $\beta=0$…