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

Part of: Elementary number theory Combinatorics Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  24 February 2021

LUKAS SPIEGELHOFER
LUKAS SPIEGELHOFER*
Affiliation:
Dept. Mathematics and Information Technology, Montanuniversität Leoben, Franz-Josef-Straße 18, 8700 Leoben, Austria. TU Wien, Wiedner Hauptstraße 8–10, 1040 Vienna, Austria. e-mail: lukas.spiegelhofer@unileoben.ac.at
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Abstract

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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 the asymptotic density

$${c_t} = \mathop {\lim }\limits_{N \to \infty } {1 \over N}|\{ 0 \le n < N:s(n + t) \ge s(n)\} |.$$
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).

MSC classification

Primary: 11A63: Radix representation; digital problems 05A20: Combinatorial inequalities
Secondary: 05A16: Asymptotic enumeration 11T71: Algebraic coding theory; cryptography
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 1 , January 2022 , pp. 139 - 161
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

Footnotes

†

The author acknowledges support by the project MuDeRa, which is a joint project between the FWF (Austrian Science Fund) and the ANR (Agence Nationale de la Recherche, France). Moreover, the author was supported by the FWF project F5502-N26, which is a part of the Special Research Program “Quasi Monte Carlo methods: Theory and Applications”.

References

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