Towards a sharp converse of Wall's theorem on arithmetic progressions

TL;DR

This paper investigates a converse to Wall's theorem on normal numbers, demonstrating that normality along sequences with densities close to one implies the original number's normality, with some limitations.

Contribution

It establishes conditions under which normality along certain subsequences implies the normality of the entire number, extending Wall's theorem.

Findings

Normality along sequences with density close to 1 implies overall normality.

Existence of non-normal numbers with normal subsequences along sequences of high density.

The result is nearly optimal, with counterexamples for densities slightly less than 1.

Abstract

Wall's theorem on arithmetic progressions says that if $0.a_1a_2a_3\dots$ is normal, then for any $k,\ell\in \mathbb{N}$, $0.a_ka_{k+\ell}a_{k+2\ell}\dots$ is also normal. We examine a converse statement and show that if $0.a_{n_1}a_{n_2}a_{n_3}\dots$ is normal for periodic increasing sequences $n_1<n_2<n_3<\dots$ of asymptotic density arbitrarily close to $1$, then $0.a_1a_2a_3\dots$ is normal. We show this is close to sharp in the sense that there are numbers $0.a_1a_2a_3\dots$ that are not normal, but for which $0.a_{n_1}a_{n_2}a_{n_3}\dots$ is normal along a large collection of sequences whose density is bounded a little away from $1$.