On the asymptotic behaviour of the correlation measure of sum-of-digits function in base 2
Résumé
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 $\mu_a$
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