Destruction of CPE-normality along deterministic sequences

TL;DR

This paper investigates how deterministic sequences affect the normality of sequences under various measures, showing that most destroy normality unless they have a very simple structure, especially for measures with positive entropy.

Contribution

It generalizes previous results by demonstrating that deterministic sets typically destroy normality for non-i.i.d. measures with CPE, except for very primitive structures called 'superficial'.

Findings

Deterministic sets with positive lower density preserve normality for i.i.d. measures.

Most deterministic sets destroy normality for measures with positive entropy and non-i.i.d. structure.

Preservation of normality can coexist with various parameters outside the CPE class.

Abstract

Let $\mu$ be a shift-invariant measure on $\Lambda^{\mathbb N}$, where $\Lambda$ is a finite or countable alphabet. We say that an infinite subset $S=\{s_1,s_2,\dots\}\subset\mathbb N$ (where $s_1<s_2<\dots$) "preserves (destroys) $\mu$-normality" if, for any $x=(x_1,x_2,\dots)\in\Lambda^{\mathbb N}$ generic for $\mu$, the sequence $x|_S=(x_{s_1},x_{s_2},\dots)$ is (is not) generic for $\mu$. It is known from Kamae and Weiss that if $\mu$ is i.i.d. then any deterministic set of positive lower density preserves $\mu$-normality. We show that deterministic sets, except ones with a very primitive structure that we call "superficial", destroy $\mu$-normality for any non-i.i.d. measure $\mu$ with completely positive entropy (CPE). This generalizes Heersink and Vandehey's result for arithmetic progressions and the Gauss measure (associated to the continued fraction transformation). We give several examples showing that, outside the class of measures with CPE, $\mu$-normality preservation can coexist with nearly any combination of three parameters: determinism of $S$ (or its lack), entropy of $\mu$ (zero or positive), and disjointness (or its lack) between $\mu$ and the measures derived from $S$.