A lower bound for Cusick’s conjecture on the digits of n + t
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in: Mathematical proceedings of the Cambridge Philosophical Society, Jahrgang 172.2022, Nr. 1, 01.2022, S. 139-161.
Publikationen: Beitrag in Fachzeitschrift › Artikel › Forschung › (peer-reviewed)
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TY - JOUR
T1 - A lower bound for Cusick’s conjecture on the digits of n + t
AU - Spiegelhofer, Lukas
N1 - Publisher Copyright: © 2021 The Author(s). Published by Cambridge University Press on behalf of Cambridge Philosophical Society.
PY - 2022/1
Y1 - 2022/1
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 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).
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 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).
KW - Cusick conjecture
KW - Hamming weight
KW - sum of digits
UR - http://www.scopus.com/inward/record.url?scp=85101594731&partnerID=8YFLogxK
U2 - 10.1017/S0305004121000153
DO - 10.1017/S0305004121000153
M3 - Article
VL - 172.2022
SP - 139
EP - 161
JO - Mathematical proceedings of the Cambridge Philosophical Society
JF - Mathematical proceedings of the Cambridge Philosophical Society
SN - 0305-0041
IS - 1
ER -
