Inverse Littlewood-Offord problems for Quasi-Norms

TL;DR

This paper investigates the structure of vector sets in high-dimensional space based on the probability that random linear combinations fall into scaled star-shaped domains, extending previous Euclidean ball results.

Contribution

It generalizes inverse Littlewood-Offord results from Euclidean balls to arbitrary star-shaped domains, broadening the understanding of vector structures under probabilistic constraints.

Findings

Extended inverse Littlewood-Offord theorems to star-shaped domains

Characterized the structure of vectors with high small ball probabilities

Connected geometric and arithmetic properties of vector sets

Abstract

Given a star-shaped domain $K\subseteq \mathbb R^d$, $n$ vectors $v_1,\dots,v_n \in \mathbb R^d$, a number $R>0$, and i.i.d. random variables $\eta_1,\dots,\eta_n$, we study the geometric and arithmetic structure of the set of vectors $V = \{v_1,\dots,v_n\}$ under the assumption that the small ball probability \[\sup_{x\in \mathbb R^d}~\mathbb P\Bigg(\sum_{j=1}^n\eta_jv_j\in x+RK\Bigg)\] does not decay too fast as $n\to \infty$. This generalises the case where $K$ is the Euclidean ball, which was previously studied by Nguyen-Vu and Tao-Vu.