Decomposing the sum-of-digits correlation measure

2024/11/12 by Bartosz Sobolewski, Sobolewski, Bartosz, Lukas Spiegelhofer +1
Mathematics · #11A63 #11T71 #FOS: Mathematics #Number Theory (math.NT) #Primary: 11B65 #Secondary: 05A20 #Statistical and numerical algorithms

paper · pdf · doi:10.48550/arxiv.2411.07779

openalex publication_date 2024/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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≥0, we consider the associated characteristic function γ_t. Our main theorem is a structural result on the decomposition of γ_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.

Related

• The binary digits of n+t
• Normal distribution of correlation measures of binary sum-of-digits functions
• Block occurrences in the binary expansion
• On the behavior of binary block-counting functions under addition
• A lower bound for Cusick's conjecture on the digits of n+t