Inverse Littlewood–Offord Problems for Quasi-norms

Abstract

Given a compact star-shaped domain K⊆ Rd, n vectors v1, … , vn∈ Rd, a number R> 0 , and i.i.d. random variables η1, ⋯ , ηn, we study the geometric and arithmetic structure of the multi-set V= { v1, … , vn} under the assumption that the concentration function (Formula Presented.) does not decay too fast as n→ ∞. 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 Rd.