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Linear fractals of the Besicovitch-Eggleston type

M. V. Pratsiovytyi, S. O. Klymchuk

Abstract

We study topological, metric and fractal properties of set of numbers $[0;1]$ with given asymptotic mean of digits in their ternary representation. We investigate connection of these numbers and numbers with a given frequency of digits.

Linear fractals of the Besicovitch-Eggleston type

Abstract

We study topological, metric and fractal properties of set of numbers with given asymptotic mean of digits in their ternary representation. We investigate connection of these numbers and numbers with a given frequency of digits.
Paper Structure (4 sections, 12 theorems, 54 equations)

This paper contains 4 sections, 12 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Relationship between digit frequencies and the asymptotic mean of digits
  3. Frequency function of a ternary digit
  4. The set of numbers with a prescribed asymptotic mean of digits

Key Result

Lemma 1

If the base $s$ of the numeral system satisfies $s>2$ and $\nu_i(x)=0$ for all $i>1$ then $r(x)$ and $\nu_1(x)$ either both do not exist or both exist and in the latter case $\nu_1(x) = r(x)$.

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • ...and 13 more