An algebraic inverse theorem for the quadratic Littlewood-Offord problem, and an application to Ramsey graphs

TL;DR

This paper extends the Littlewood-Offord problem to quadratic polynomials, establishing that high concentration implies low-rank quadratic structure, and applies this to solve a problem in Ramsey graph theory.

Contribution

It provides an algebraic inverse theorem for quadratic Littlewood-Offord problems and applies it to a Ramsey graph problem, advancing understanding of polynomial concentration.

Findings

High concentration implies quadratic form has low rank

Quadratic polynomials with large point probabilities are close to low-rank forms

Application to asymptotic Ramsey graph properties

Abstract

Consider a quadratic polynomial $f\left(\xi_{1},\dots,\xi_{n}\right)$ of independent Bernoulli random variables. What can be said about the concentration of $f$ on any single value? This generalises the classical Littlewood--Offord problem, which asks the same question for linear polynomials. As in the linear case, it is known that the point probabilities of $f$ can be as large as about $1/\sqrt{n}$, but still poorly understood is the "inverse" question of characterising the algebraic and arithmetic features $f$ must have if it has point probabilities comparable to this bound. In this paper we prove some results of an algebraic flavour, showing that if $f$ has point probabilities much larger than $1/n$ then it must be close to a quadratic form with low rank. We also give an application to Ramsey graphs, asymptotically answering a question of Kwan, Sudakov and Tran.