发表机构
Institute of Science and Technology Austria (ISTA)（奥地利科学技术研究所（ISTA））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文展示GPT-6 Pro发现的论证，给出多项式Littlewood-Offord问题的最优界，改进先前结果并解决相关猜想。
AI 中文摘要
我们展示了一个由GPT-6 Pro发现的论证，该论证给出了多项式Littlewood-Offord问题的最优界。具体而言，设$F$是一个次数为$d$的多线性多项式，包含$r$个涉及不相交变量集合的次数为$d$的单项式。那么，对于独立同分布的Rademacher随机变量$\xi_1, \ldots, \xi_n$，我们有$\mathbb{P}[F(\xi_1, \ldots, \xi_n) = 0] = O_d(r^{-1/2})$。这改进了Meka、O. Nguyen和Vu先前得到的$(\log r)^{O_d(1)} r^{-1/2}$的界，并解决了归因于H. Nguyen和Vu的一个猜想。证明的关键部分是对有界次数有理函数的总影响的一个估计，该估计解决了Kothari、Kovacs-Deak、Wang和Yang最近的一个猜想。
英文摘要
We present an exposition of an argument, discovered by GPT-6 Pro, that gives an optimal bound for the polynomial Littlewood-Offord problem. Namely, let $F$ be a degree-$d$ multilinear polynomial that contains $r$ degree-$d$ monomials involving disjoint sets of variables. Then, for i.i.d. Rademacher random variables $ξ_1, \ldots, ξ_n$, we have $\mathbb{P}[F(ξ_1, \ldots, ξ_n) = 0] = O_d(r^{-1/2})$. This improves upon the previous bound of $(\log r)^{O_d(1)} r^{-1/2}$ due to Meka, O. Nguyen, and Vu, and resolves a conjecture attributed to H. Nguyen and Vu. The key part of the proof is an estimate for the total influence of bounded-degree rational functions, which resolves a recent conjecture of Kothari, Kovacs-Deak, Wang, and Yang.
Comments9 pages
