# Combinatorial anti-concentration inequalities, with applications

## TL;DR

This paper establishes new anti-concentration inequalities for functions of independent Bernoulli variables, extending classical results and applying them to subgraph counts in random graphs.

## Contribution

It proves novel Poisson-type and polynomial anti-concentration inequalities, extending classical theorems and addressing conjectures in the field.

## Key findings

- Bounds of 1/e + o(1) for certain polynomials' point probabilities
- Extended Erdős-Littlewood-Offord theorem for nonnegative coefficient polynomials
- New anti-concentration bounds for subgraph counts in random graphs

## Abstract

We prove several different anti-concentration inequalities for functions of independent Bernoulli-distributed random variables. First, motivated by a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn, we prove some "Poisson-type" anti-concentration theorems that give bounds of the form 1/e + o(1) for the point probabilities of certain polynomials. Second, we prove an anti-concentration inequality for polynomials with nonnegative coefficients which extends the classical Erd\H{o}s-Littlewood-Offord theorem and improves a theorem of Meka, Nguyen and Vu for polynomials of this type. As an application, we prove some new anti-concentration bounds for subgraph counts in random graphs.