Sharp extremal asymptotics for Cusick's sum-of-digits bias at fixed Hamming weight

Abstract

Let $s_2(n)$ be the binary sum-of-digits function and let $c_t$ be the natural density of the integers $n\ge0$ for which $s_2(n+t)\ge s_2(n)$. Earlier work of the author proved the universal exponential bound $$c_t-\frac12\ge 2^{-2s_2(t)-1},$$ thereby resolving Cusick's conjecture for every $t$. This estimate, however, does not reflect the true size of the smallest possible bias at a given large Hamming weight. In this paper, we determine this extremal scale sharply: $$\inf_{s_2(t)=k}\left(c_t-\frac12\right) \sim \frac{1}{2\sqrtπ} \left(\frac{\log_2 k}{k}\right)^{3/2} \qquad(k\to\infty).$$ Thus the optimal fixed-weight gap is polynomial-logarithmic rather than exponential, with the explicit sharp leading constant $1/(2\sqrtπ)$. The proof combines the five-cumulant Edgeworth expansion of Spiegelhofer and Wallner with a new extremal rigidity mechanism for near-extremal binary block patterns. We also prove a stability theorem for asymptotic extremizers and give a separate shadow-energy interpretation of the same constant.