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 $\epsilon >0$ we have \[ c_t+c_{t'}>1-\varepsilon \] if the binary expansion of $t$ contains enough blocks of consecutive $1$s (depending on $\epsilon$), where $t^{\prime }=3\cdot 2^{\lambda }-t$ and $\lambda$ is chosen such that $2^{\lambda }\le t<2^{\lambda +1}$.

7 pages

Cited by in corpus (1)

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