Approaching Cusick's Conjecture on the Sum-of-digits Function
Lukas Spiegelhofer · arXiv (Cornell University) · 2019
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 \[ c_t=\lim_{N\rightarrow\infty}\frac 1N\left\lvert\{n1/2. \] 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 $2^λ\leq t<2^{λ+1}$.