Algebraic aspects of the polynomial Littlewood-Offord problem

TL;DR

This paper investigates the concentration of polynomial functions of independent Rademacher variables, improving existing bounds and revealing algebraic structures that influence probability concentration, with implications in number theory.

Contribution

It provides new bounds on polynomial concentration probabilities and identifies algebraic structures that affect these bounds, correcting previous conjectures.

Findings

Improved bounds on point probabilities for polynomial functions

Identification of algebraic structures influencing concentration

Disproof of Costello's original conjecture on multilinear forms

Abstract

Consider a degree-$d$ polynomial $f(\xi_1,\dots,\xi_n)$ of independent Rademacher random variables $\xi_1,\dots,\xi_n$. To what extent can $f(\xi_1,\dots,\xi_n)$ concentrate on a single point? This is the so-called polynomial Littlewood-Offord problem. A nearly optimal bound was proved by Meka, Nguyen and Vu: the point probabilities are always at most about $1/\sqrt n$, unless $f$ is "close to the zero polynomial" (having only $o(n^d)$ nonzero coefficients). In this paper we prove several results supporting the general philosophy that the Meka-Nguyen-Vu bound can be significantly improved unless $f$ is "close to a polynomial with special algebraic structure", drawing some comparisons to phenomena in analytic number theory. In particular, one of our results is a corrected version of a conjecture of Costello on multilinear forms (in an appendix with Ashwin Sah and Mehtaab Sawhney, we disprove Costello's original conjecture).