Normal distribution of correlation measures of binary sum-of-digits functions

TL;DR

This paper proves that correlation measures of binary sum-of-digits functions follow a normal distribution under broad conditions, extending previous results limited to Bernoulli measures, and establishing a central limit theorem for these measures.

Contribution

It generalizes the central limit theorem for correlation measures of sum-of-digits functions to all shift-invariant ergodic measures, beyond the symmetric Bernoulli case.

Findings

Correlation measures satisfy a central limit theorem under ergodic measures.

Normal distribution emerges for the correlation measures in the asymptotic limit.

Results extend previous work from Bernoulli measures to all shift-invariant ergodic measures.

Abstract

In this paper we study correlation measures introduced in \cite{emme_asymptotic_2017}. Denote by $\mu_a(d)$ the asymptotic density of the set $\mathcal{E}_{a,d}=\{n \in \mathbb{N}, \ s_2(n+a)-s_2(n)=d\}$ (where $s_2$ is the sum-of-digits function in base 2). Then, for any point $X$ in $\{0,1\}^\mathbb{N}$, define the integer sequence $\left(a_X (n)\right)_{n\in \mathbb{N}}$ such that the binary decomposition of $a_X (n) $ is the prefix of length $n$ of $X$. We prove that for \textit{any} shift-invariant ergodic probability measure $\nu$ on $\{0,1\}^\mathbb{N}$, the sequence $\left(\mu_{a_X(n)}\right)_{n \in \mathbb{N}}$ satisfies a central limit theorem. This result was proven in the case where $\nu$ is the symmetric Bernoulli measure in \cite{emme_central_2018}.