Decomposing the sum-of-digits correlation measure

Let s(n) denote the number of ones in the binary expansion of the nonnegative integer n. How does s behave under addition of a constant t? In order to study the differences s(n + t) − s(n), for all n ≥ 0, we consider the associated characteristic function γt. Our main theorem is a structural result on the decomposition of γt into a sum of components. We also study in detail the case that t contains at most two blocks of consecutive 1s. The results in this paper are motivated by Cusick’s conjecture on the sum-of-digits function. This conjecture is concerned with the central tendency of the corresponding probability distributions, and is still unsolved.