A lower bound for Cusick's conjecture on the digits of n+t

2019/10/29 by Lukas Spiegelhofer, Spiegelhofer, Lukas
Mathematics · #05A20 #11T71 #Advanced Mathematical Identities #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #Primary: 11A63 #Secondary: 05A16

paper · pdf · doi:10.48550/arxiv.1910.13170

openalex publication_date 2019/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let s be the sum-of-digits function in base 2, which returns the number of \mathtt 1s in the base-2 expansion of a nonnegative integer. For a nonnegative integer t, define the asymptotic density \[ c_t=lim_{N→ ∞} \frac 1N|\0≤ n1/2. We have the elementary bound 01/2-ε as soon as t contains sufficiently many blocks of \mathtt 1s in its binary expansion. In the proof, we provide estimates for the moments of an associated probability distribution; this extends the study initiated by Emme and Prikhod'ko (2017) and pursued by Emme and Hubert (2018).

Citations

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

Related

• The binary digits of n+t
• Approaching Cusick's Conjecture on the Sum-of-digits Function
• Block occurrences in the binary expansion
• Decomposing the sum-of-digits correlation measure
• Somme des chiffres et changement de base