Resolution of the quadratic Littlewood--Offord problem

TL;DR

This paper establishes an optimal bound on the probability that a quadratic polynomial of independent Rademacher variables concentrates on a single value, advancing understanding of quadratic anticoncentration and resolving related conjectures.

Contribution

It provides an essentially optimal bound for quadratic Littlewood--Offord problems, confirming a conjecture by Nguyen and Vu, and introduces a novel inductive decoupling technique.

Findings

Bound on quadratic polynomial concentration: O(1/√m)

Extension of results to arbitrary distributions of variables

Resolution of a conjecture on graph inducibility

Abstract

Consider a quadratic polynomial $Q(\xi_{1},\dots,\xi_{n})$ of independent Rademacher random variables $\xi_{1},\dots,\xi_{n}$. To what extent can $Q(\xi_{1},\dots,\xi_{n})$ concentrate on a single value? This quadratic version of the classical Littlewood--Offord problem was popularised by Costello, Tao and Vu in their study of symmetric random matrices. In this paper, we obtain an essentially optimal bound for this problem, as conjectured by Nguyen and Vu. Specifically, if $Q(\xi_{1},\dots,\xi_{n})$ "robustly depends on at least $m$ of the $\xi_{i}$" in the sense that there is no way to pin down the value of $Q(\xi_{1},\dots,\xi_{n})$ by fixing values for fewer than $m$ of the variables $\xi_{i}$, then we have $\Pr[Q(\xi_{1},\dots,\xi_{n})=0]\le O(1/\sqrt{m})$. This also implies a similar result in the case where $\xi_{1},\dots,\xi_{n}$ have arbitrary distributions. Our proof combines a number of ideas that may be of independent interest, including an inductive decoupling scheme that reduces quadratic anticoncentration problems to high-dimensional linear anticoncentration problems. Also, one application of our main result is the resolution of a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn related to graph inducibility.