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

Jordan Emme (1, 2) , Alexander Prikhod'Ko (3)

1 AMU - Aix Marseille Université
2 I2M - Institut de Mathématiques de Marseille
3 MIPT - Moscow Institute of Physics and Technology [Moscow]

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 $\mu_a$

Mots clés: Number Theory; Combinatorics; Probabilities

Domaines: Combinatoire [math.CO]; Probabilités [math.PR]; Théorie des nombres [math.NT]

Jordan Emme, Alexander Prikhod'Ko. On the asymptotic behaviour of the correlation measure of sum-of-digits function in base 2. Integers : Electronic Journal of Combinatorial Number Theory, 2017, 17, pp.A58. ⟨hal-01138865v2⟩