固定汉明重量下Cusick数位和偏差的尖锐极值渐近分析
Sharp extremal asymptotics for Cusick's sum-of-digits bias at fixed Hamming weight
AI总结：
本文针对二进制数位和函数，解决Cusick猜想后进一步确定固定汉明重量下偏差的尖锐极值渐近尺度，证明最优偏差为多项式对数级并给出首项常数，结合相关方法完成证明并补充稳定性定理与常数的阴影能量解释。
AI中文摘要：
设$s_2(n)$为二进制数位和函数，$c_t$为满足$s_2(n+t)\boldsymbol{\text{≥}}s_2(n)$的非负整数$n$的自然密度。作者早期工作证明了通用指数界$c_t-\frac{1}{2}\boldsymbol{\text{≥}}2^{-2s_2(t)-1}$，从而解决了对所有$t$的Cusick猜想。然而，该估计未反映给定大汉明重量下最小偏差的真实规模。本文我们精确确定该极值尺度：当$k\to\boldsymbol{\text{∞}}$时，$\boldsymbol{\text{inf}}_{s_2(t)=k}\bigl(c_t-\frac{1}{2}\bigr)\boldsymbol{\text{∼}}\frac{1}{2\boldsymbol{\text{√π}}}\bigl(\frac{\boldsymbol{\text{log}}_2k}{k}\bigr)^{3/2}$。因此最优固定权重偏差是多项式对数级而非指数级，且具有明确的尖锐首项常数$\frac{1}{2\boldsymbol{\text{√π}}}$。证明结合了Spiegelhofer和Wallner的五累积量Edgeworth展开，以及针对近极值二进制块模式的新型极值刚性机制。我们还证明了渐近极值的稳定性定理，并给出同一常数的独立阴影能量解释。
英文摘要：
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.
