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arXiv · 1904.08646

Approaching Cusick's conjecture on the sum-of-digits function

Lukas Spiegelhofer
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Abstract

Cusick's conjecture on the binary sum of digits $s(n)$ of a nonnegative integer $n$ states the following: for all nonnegative integers $t$ we have \[ c_t=\lim_{N\rightarrow\infty}\frac 1N\left\lvert\{n 1/2. \] We prove that for given $\varepsilon>0$ we have \[ c_t+c_{t'}>1-\varepsilon \] if the binary expansion of $t$ contains enough blocks of consecutive $\mathtt 1$s (depending on $\varepsilon$), where $t'=3\cdot 2^λ-t$ and $λ$ is chosen such that $2^λ\leq t<2^{λ+1}$.

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Lukas Spiegelhofer. 2019-04-18. Approaching Cusick's conjecture on the sum-of-digits function. https://arxiv.org/abs/1904.08646

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