## Abstract

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 the asymptotic density 'Equation Presented' 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 ≤ ß < 1, valid for all t, is known. In this paper, we prove that ct > 1/2 - ∈ 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 language | English |
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Pages (from-to) | 139-161 |

Number of pages | 23 |

Journal | Mathematical proceedings of the Cambridge Philosophical Society |

Volume | 172.2022 |

Issue number | 1 |

Early online date | 24 Feb 2021 |

DOIs | |

Publication status | Published - Jan 2022 |

### Bibliographical note

Publisher Copyright: © 2021 The Author(s). Published by Cambridge University Press on behalf of Cambridge Philosophical Society.## Keywords

- Cusick conjecture
- Hamming weight
- sum of digits