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

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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 0<ct<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).

Original languageEnglish
Number of pages23
JournalMathematical proceedings of the Cambridge Philosophical Society
Publication statusPublished - 2021


  • Cusick conjecture
  • Hamming weight
  • sum of digits

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