Decomposing the sum-of-digits correlation measure

TL;DR

This paper analyzes how the binary digit sum function behaves under addition by a constant, providing a structural decomposition of its characteristic function and exploring special cases related to Cusick's conjecture.

Contribution

It introduces a novel decomposition of the characteristic function of the sum-of-digits difference, advancing understanding of its structure and special cases.

Findings

Decomposition of the characteristic function into components

Detailed analysis for t with at most two blocks of 1s

Insights relevant to Cusick's conjecture

Abstract

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 $n\ge0$, 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 \emph{components}. We also study in detail the case that $t$ contains at most two blocks of consecutive $1$s. The results in this paper are motivated by \emph{Cusick's conjecture} on the sum-of-digits function. This conjecture is concerned with the \emph{central tendency} of the corresponding probability distributions, and is still unsolved.