Approaching Cusick's Conjecture on the Sum-of-digits Function

2019/04/18 by Lukas Spiegelhofer, Spiegelhofer, Lukas · 1 citation
Computer Science · Mathematics · #05A20 #11A63 #11B50 #Analytic Number Theory Research #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1904.08646

openalex publication_date 2019/04/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Cusick's conjecture on the binary sum of digits s(n) of a nonnegative integer n states the following: for all nonnegative integers t we have c_t=lim_N→∞\frac 1N|\n1/2. We prove that for given ε>0 we have c_t+c_t'gt;1-ε if the binary expansion of t contains enough blocks of consecutive \mathtt 1s (depending on ε), where t'=3⋅ 2^λ-t and λ is chosen such that 2^λ≤ t<2^λ+1.

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