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 $\xi_1, \ldots, \xi_n$, we have $\mathbb{P}[F(\xi_1, \ldots, \xi_n) = 0] = O_d(r^{-1/2})$. 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.
Let $q=p^f$, where $p$ is prime. We prove that every $q$-ary linear code meeting the Griesmer bound, with positive dimension and minimum distance $d\ge1$, is $p^{\max\{0,\nu_p(d)-f+1\}}$-divisible, as asserted in Ward's conjecture. Arbitrary projective column multiplicities are allowed. The geometric ingredient is a bo...
A degree-$d$ polynomial source is the output of a polynomial map of degree at most $d$ over $\mathbb{F}_2$ on arbitrarily many uniform random bits. Khodabandeh and Shinkar (FOCS'26) proved that $\mathrm{Ber}(1/3)^{\otimes N}$ has statistical distance $1-o(1)$ from every constant-degree polynomial source and conjectured...
The degree-distortion tradeoff for polynomial approximation of the $d$-dimensional cross-polytope $B_1^d$ is determined, and degree $\Theta(d)$ is necessary and sufficient for constant distortion.
We prove that, for every constant $\varepsilon>0$ and every function $f:\Sigma^n\to\{0, 1\}$, there is a coalition of $O(n/\sqrt{\log n})$ coordinates and a target output $b\in\{0, 1\}$ such that, after the remaining coordinates are sampled uniformly and independently, the coalition can choose its values to make the ou...
We show that the worst-case combinatorial discrepancy of $n$ points in the plane with respect to axis-parallel rectangles is $\Theta(\log^{3/2}n)$. The known bounds were $\Omega(\log n)$ and $O(\log^{3/2}n)$; we prove the matching lower bound. It holds for random point sets: for every $A>0$, there is a constant $c_A>0$...
We show that for all positive, nonincreasing, convex functions $f : (0, \infty) \to (0,\infty)$ and for any collection of $n$ points $x_i \in \mathbb{R}$ with $x_1<x_2<\ldots<x_n$ and $n$ even, the Pfaffian of the skew matrix defined by $f(x_j - x_i)$ above the diagonal is positive. More generally, we establish coeffic...
William T. Johnson· 0 citations
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