Inverse Littlewood-Offord problems for Quasi-Norms

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 $$\underset{x\in {\mathbb{R}}^d}{\sup }\mathbb{P}(\sum _{j=1}^n{\eta }_j{v}_j\in x+RK)$$ does not decay too fast as $n\to \mathrm{\infty }$. This generalises the case where $K$ is the Euclidean ball, which was previously studied by Nguyen-Vu and Tao-Vu.