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

Let $s\mathrm{\_}2(x)$ denote the number of digits “$1$” in a binary expansion of any $x\in \mathbb{N}$. We study the mean distribution $\mu \mathrm{\_}a$ of the quantity $s\mathrm{\_}2(x+a)-s\mathrm{\_}2(x)$ for a fixed positive integer $a$.It is shown that solutions of the equation$$s\mathrm{\_}2(x+a)-s\mathrm{\_}2(x)=d$$are uniquely identified by a finite set of prefixes in $\{0,1{\}}^{\ast }$, 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 $\mu \mathrm{\_}a$