On preservation of normality and determinism under arithmetic operations

TL;DR

This paper develops an ergodic approach to study how arithmetic operations affect normality and determinism in numbers, generalizing Rauzy's theorem to ergodic toral endomorphisms and exploring the effects of different measures.

Contribution

It introduces a new ergodic framework for analyzing normality and determinism under arithmetic operations, extending Rauzy's theorem to broader dynamical systems.

Findings

Generalizes Rauzy's theorem to ergodic toral endomorphisms.

Shows that Rauzy phenomena do not hold under certain non-uniform Bernoulli measures.

Provides examples where Rauzy-type results fail under multiplication instead of addition.

Abstract

In this paper we develop a general ergodic approach which reveals the underpinnings of the effect of arithmetic operations involving normal and deterministic numbers. This allows us to recast in new light and amplify the result of Rauzy, which states that a number $y$ is deterministic if and only if $x+y$ is normal for every normal number $x$. Our approach is based on the notions of lower and upper entropy of a point in a topological dynamical system. The ergodic approach to Rauzy theorem naturally leads to the study of various aspects of normality and determinism in the general framework of dynamics of endomorphisms of compact metric groups. In particular, we generalize Rauzy theorem to ergodic toral endomorphisms. Also, we show that the phenomena described by Rauzy do not occur when one replaces the base $2$ normality associated with the $(\frac12,\frac12)$-Bernoulli measure by the variant of normality associated with a $(p,1-p)$-Bernoulli measure, where $p\neq\frac12$. Finally, we present some rather nontrivial examples which show that Rauzy-type results are not valid when addition is replaced by multiplication.