Anti-concentration for polynomials of independent random variables

TL;DR

This paper establishes anti-concentration bounds for polynomials of independent random variables, extending classical results and applying them to complexity theory and random graph analysis.

Contribution

It generalizes anti-concentration results to higher-degree polynomials and demonstrates their applications in complexity bounds and graph theory.

Findings

Proves near optimal lower bounds for Parity computation.

Derives anti-concentration results for graph copy counts.

Extends Littlewood-Offord type results to arbitrary degree polynomials.

Abstract

We prove anti-concentration results for polynomials of independent random variables with arbitrary degree. Our results extend the classical Littlewood-Offord result for linear polynomials, and improve several earlier estimates. We discuss applications in two different areas. In complexity theory, we prove near optimal lower bounds for computing the Parity, addressing a challenge in complexity theory posed by Razborov and Viola, and also address a problem concerning OR functions. In random graph theory, we derive a general anti-concentration result on the number of copies of a fixed graph in a random graph.