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

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