Transformations and functions that preserve the asymptotic mean of digits in the ternary representation of a number

TL;DR

This paper investigates transformations of the interval [0,1) that preserve the asymptotic mean of digits in the s-adic representation, providing necessary and sufficient conditions for such transformations.

Contribution

It characterizes the class of functions that preserve the asymptotic digit mean in s-adic representations, offering a complete criterion for their identification.

Findings

Identifies conditions for transformations to preserve digit mean

Provides a full characterization of such transformations

Advances understanding of digit distribution invariance

Abstract

In the paper we study transformations of the interval $[0;1)$ and functions that preserve the asymptotic mean $r$ of the digits in the $s$--adic representation of a number $x$, $$r(x)=\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits^{n}_{i=1}\alpha_i(x)$$ specify the necessary and sufficient conditions for a transformation to belong to this class