Small ball probability, Inverse theorems, and applications

TL;DR

This paper reviews recent advances in understanding the small ball probability for random sums, characterizing the structure of sets that lead to high probabilities and discussing applications in solving open problems.

Contribution

It provides a comprehensive overview of recent inverse theorems related to small ball probabilities and their applications across various fields.

Findings

Characterization of sets with high small ball probability

Progress on open problems in probability and related areas

Connections between structure of sets and probability estimates

Abstract

Let $\xi$ be a real random variable with mean zero and variance one and $A={a_1,...,a_n}$ be a multi-set in $\R^d$. The random sum $$S_A := a_1 \xi_1 + ... + a_n \xi_n $$ where $\xi_i$ are iid copies of $\xi$ is of fundamental importance in probability and its applications. We discuss the small ball problem, the aim of which is to estimate the maximum probability that $S_A$ belongs to a ball with given small radius, following the discovery made by Littlewood-Offord and Erdos almost 70 years ago. We will mainly focus on recent developments that characterize the structure of those sets $A$ where the small ball probability is relatively large. Applications of these results include full solutions or significant progresses of many open problems in different areas.