A first-exit proof of Cusick's sum-of-digits conjecture

We prove Cusick’s conjecture on the binary sum-of-digits function. More precisely, for every integer $t\ge 1$ we show that $$c_t:=\underset{N\to \mathrm{\infty }}{\lim }\frac{1}{N}\mathrm{\#}\{0\le n<N:s_2(n+t)\ge s_2(n)\}>\frac{1}{2},$$ and in fact obtain the explicit bound $$c_t\ge \frac{1}{2}+2^{-2s_2(t)-1},$$ where $s_2(m)$ denotes the number of ones in the binary expansion of $m$. The proof is based on an exact deconvolution which replaces the distribution of $s_2(n+t)-s_2(n)$ by a finite stopped random-walk law. The required bias is then proved through first-exit medians for principal subsequence ideals.