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

Abstract: 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:=\lim_{N\to\infty}\frac{1}{N} #{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.