A continuous variant of the inverse Littlewood-Offord problem for quadratic forms

TL;DR

This paper investigates the concentration behavior of quadratic forms, revealing that high concentration implies the coefficients have a specific additive and algebraic structure, extending inverse Littlewood-Offord results.

Contribution

It introduces a continuous variant of the inverse Littlewood-Offord problem for quadratic forms, characterizing coefficient structures from concentration phenomena.

Findings

Quadratic forms with high concentration have coefficients approximable by additive and algebraic structures.

The study extends inverse Littlewood-Offord theory from linear to quadratic forms.

Provides a structural description of coefficients based on concentration properties.

Abstract

Motivated by the inverse Littlewood-Offord problem for linear forms, we study the concentration of quadratic forms. We show that if this form concentrates on a small ball with high probability, then the coefficients can be approximated by a sum of additive and algebraic structures.