TY - JOUR

T1 - A lower bound for Cusick’s conjecture on the digits of n + t

AU - Spiegelhofer, Lukas

PY - 2021

Y1 - 2021

N2 - Let s be the sum-of-digits function in base 2, which returns the number of 1s in the base-2 expansion of a nonnegative integer. For a nonnegative integer t, define ct as the asymptotic density of nonnegative integers n such that s(n+t)≥s(n). T. W. Cusick conjectured that ct>1/2.  We have the elementary bound 0t<1; however, no bound of the form 0<α≤ct or ct≤β <1, valid for all t, is known. In this paper, we prove that ct>1/2−ε as soon as t contains sufficiently many blocks of 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).

AB - Let s be the sum-of-digits function in base 2, which returns the number of 1s in the base-2 expansion of a nonnegative integer. For a nonnegative integer t, define ct as the asymptotic density of nonnegative integers n such that s(n+t)≥s(n). T. W. Cusick conjectured that ct>1/2.  We have the elementary bound 0t<1; however, no bound of the form 0<α≤ct or ct≤β <1, valid for all t, is known. In this paper, we prove that ct>1/2−ε as soon as t contains sufficiently many blocks of 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).

KW - Cusick conjecture

KW - Hamming weight

KW - sum of digits

M3 - Article

VL - 2021

JO - Mathematical proceedings of the Cambridge Philosophical Society

JF - Mathematical proceedings of the Cambridge Philosophical Society

SN - 0305-0041

ER -