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

Abstract

We prove Cusick&#39;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.