Linear fractals of the Besicovitch-Eggleston type

TL;DR

This paper explores the topological, metric, and fractal characteristics of subsets of [0,1] defined by specific digit frequency constraints in their ternary expansions, revealing their intricate structure and relationships.

Contribution

It introduces a detailed analysis of fractal sets based on digit frequency in ternary representations, connecting asymptotic digit means with fractal properties.

Findings

Characterization of fractal properties of digit frequency sets

Connections between asymptotic digit means and number distributions

Insights into the structure of sets with prescribed digit frequencies

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.