Inverse Littlewood-Offord problems for Quasi-Norms

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 \[\sup_{x\in \mathbb R^d}~\mathbb P\Bigg(\sum_{j=1}^nη_jv_j\in x+RK\Bigg)\] 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.