文摘
Given a compact star-shaped domain \(K\subseteq \mathbb {R}^d\), n vectors \(v_1,\ldots ,v_n \in \mathbb {R}^d\), a number \(R>0\), and i.i.d. random variables \(\eta _1,\dots ,\eta _n\), we study the geometric and arithmetic structure of the multi-set \(V = \{v_1,\ldots ,v_n\}\) under the assumption that the concentration function $$\begin{aligned} \sup _{x\in \mathbb {R}^d}~\mathbb {P}\Big (\sum _{j=1}^n\eta _j v_j\in x+RK\Big ) \end{aligned}$$does not decay too fast as \(n\rightarrow \infty \). This generalises the case where K is the Euclidean ball, which was previously studied in Nguyen and Vu (Adv Math 226(6):5298–5319, 2011) and Tao and Vu (Combinatorica 32(3):363–372, 2012), to the non-Euclidean settings, that is, to general norms and quasi-norms in \(\mathbb {R}^d\).KeywordsConcentration functionQuasi-normInverse Littlewood-Offord problemsEditor in Charge: János Pach
