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

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 $\mu_a$