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

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 \[ c_t=\lim_{N\rightarrow\infty}\frac 1N\left\lvert\{n<N:s(n+t)\geq s(n)\}\right\rvert>1/2. \] 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 $t'=3\cdot 2^\lambda-t$ and $\lambda$ is chosen such that $2^\lambda\leq t<2^{\lambda+1}$.