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arXiv · 1910.13170

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

Lukas Spiegelhofer
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Abstract

Let $s$ be the sum-of-digits function in base $2$, which returns the number of $\mathtt 1$s in the base-2 expansion of a nonnegative integer. For a nonnegative integer $t$, define the asymptotic density \[ c_t=\lim_{N\rightarrow \infty} \frac 1N\bigl\lvert\{0\leq n 1/2$. We have the elementary bound $0 1/2-\varepsilon$ as soon as $t$ contains sufficiently many blocks of $\mathtt 1$s 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).

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Lukas Spiegelhofer. 2019-11-15. A lower bound for Cusick's conjecture on the digits of n+t. https://arxiv.org/abs/1910.13170

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