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Asymptotic mean of digits of the $Q_s$-representation of the fractional part of a real number and related problems of fractal geometry and fractal analysis

M. V. Pratsiovytyi, S. O. Klymchuk

Abstract

We introduce a concept of asymptotic mean of digits (symbols) in the $Q_s$-representation of a real number, that is a generalization of the $s$-adic representation and have a self-similar geometry. We discuss its relationship with the frequencies of digits and formulate problems related to the concept. We study the topological, metric, and fractal properties of the set of real numbers that have no asymptotic mean of $Q_s$-symbols. Also we study topological, metric and fractal properties of the sets of real numbers that have asymptotic mean of $Q_3$-symbols which is equal to value of digit frequency of number.

Asymptotic mean of digits of the $Q_s$-representation of the fractional part of a real number and related problems of fractal geometry and fractal analysis

Abstract

We introduce a concept of asymptotic mean of digits (symbols) in the -representation of a real number, that is a generalization of the -adic representation and have a self-similar geometry. We discuss its relationship with the frequencies of digits and formulate problems related to the concept. We study the topological, metric, and fractal properties of the set of real numbers that have no asymptotic mean of -symbols. Also we study topological, metric and fractal properties of the sets of real numbers that have asymptotic mean of -symbols which is equal to value of digit frequency of number.
Paper Structure (4 sections, 4 theorems, 32 equations)

This paper contains 4 sections, 4 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. The object of our study and the related problems.
  3. The relationship between the asymptotic mean of digits and their frequencies. The asymptotic mean of digits function
  4. Normality of numbers and the role of normality in the theory of singular functions

Key Result

Theorem 1

The set $S$ is a continuous, everywhere dense and everywhere discontinuous set of zero Lebesgue measure. The set $S$ is superfractal, that is, its Hausdorff–Besicovitch dimension $\alpha_0(S)$ equals $1$.

Theorems & Definitions (9)

  • Definition 1
  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof