Optimal bound for the polynomial Littlewood-Offord problem

Source abstract

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.