Decomposing the sum-of-digits correlation measure

Abstract

Let $s\left(n\right)$ 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$s(n+t)-s\left(n\right),$ for all $n\ge 0$, we consider the associated characteristic function ${\gamma }_t$. Our main theorem is a structural result on the decomposition of ${\gamma }_t$ 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.