Bilinear and Quadratic Variants on the Littlewood-Offord Problem

TL;DR

This paper investigates the maximum concentration of polynomial functions, specifically bilinear and quadratic forms, of independent Bernoulli variables, extending classical Littlewood-Offord results to these nonlinear cases.

Contribution

It characterizes the concentration bounds for bilinear and quadratic forms, identifying near-tight conditions under which high concentration occurs, thus extending Littlewood-Offord theory.

Findings

Bilinear forms with high concentration are nearly degenerate.

Quadratic forms with many nonzero coefficients have concentration at most about n^{-1/2}.

Results are nearly optimal and extend classical Littlewood-Offord bounds.

Abstract

If f(x_1, x_2, ..., x_n) is a polynomial dependent on a large number of independent Bernoulli random variables, what can be said about the maximum concentration of f on any single value? For linear polynomials, this reduces to one version of the classical Littlewood-Offord problem: Given nonzero constants a_1 through a_n, what is the maximum number of sums of the form +/- a_1 +/- a_2 +/-... +/- a_n which take on any single value? Here we consider the case where f is either a bilinear form or a quadratic form. For the bilinear case, we show that the only forms having concentration significantly larger than n^{-1} are those which are in a certain sense very close to being degenerate. For the quadratic case, we show that no form having many nonzero coefficients has concentration significantly larger than n^{-1/2}. In both cases the results are nearly tight.