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On the Asymptotic Behavior of Density of Sets Defined by Sum-of-digits Function in Base 2

Let s2(x) denote the number of occurrences of the digit “1” in the binary expansion of x in N. We study the mean distribution μa of the quantity s2(x + a) s2(x) for a fixed positive integer a. It is shown that solutions of the equation s2(x + a) s2(x) = d are uniquely identified by a finite set of prefixes in {0, 1}⇤, and that the probability distribution of di↵erences d is given by an infinite product of matrices whose coefficients are operators of l1(Z). Then, denoting by l(a) the number of occurrences of the pattern “01” in the binary expansion of a, we give the asymptotic behavior of this probability distribution as l(a) goes to infinity, as well as estimates of the variance of the probability measure μa.