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

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}.