On the asymptotic behaviour of the correlation measure of sum-of-digits function in base 2

TL;DR

This paper investigates the asymptotic behavior of the correlation measure of the sum-of-digits function in base 2, revealing how the distribution of digit sum differences evolves with the pattern complexity of the fixed integer.

Contribution

It introduces a novel matrix product approach to analyze the distribution of sum-of-digits differences and characterizes its asymptotic behavior based on binary pattern complexity.

Findings

Distribution of differences is given by an infinite product of matrices.

Asymptotic behavior depends on the number of '01' patterns in the binary expansion.

Provides estimates for the variance of the distribution.

Abstract

Let $s\_2(x)$ denote the number of digits "$1$" in a binary expansion of any $x \in \mathbb{N}$. We study the mean distribution $\mu\_a$ of the quantity $s\_2(x+a)-s\_2(x)$ for a fixed positive integer $a$.It is shown that solutions of the equation$$ s\_2(x+a)-s\_2(x)= d $$are uniquely identified by a finite set of prefixes in $\{0,1\}^*$, and that the probability distribution of differences $d$ is given by an infinite product of matrices whose coefficients are operators of $l^1(\mathbb{Z})$.Then, denoting by $l(a)$ the number of patterns "$01$" in the binary expansion of $a$, we give the asymptotic behaviour of this probability distribution as $l(a)$ goes to infinity as well as estimates of the variance of the probability measure $\mu\_a$