Inverse Littlewood-Offord problems for Quasi-Norms

2015/10/14 by Omer Friedland, Friedland, Omer, Ohad Giladi +3
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1510.03937

arxiv created 2015/10/14 · arxiv updated 2015/10/15

Abstract

Given a star-shaped domain K⊆ \mathbb R^d, n vectors v₁,…,v_n ∈ \mathbb R^d, a number R>0, and i.i.d. random variables η₁,…,η_n, we study the geometric and arithmetic structure of the set of vectors V = \v₁,…,v_n\ under the assumption that the small ball probability sup_{x∈ \mathbb R^d}~\mathbb P(∑_j=1ⁿη_jv_j∈ x+RK) does not decay too fast as n→ ∞. This generalises the case where K is the Euclidean ball, which was previously studied by Nguyen-Vu and Tao-Vu.

Related

• A nonuniform Littlewood-Offord inequality for all norms
• The Littlewood-Offord problem in high dimensions and a conjecture of Frankl and Füredi
• Small ball probability, Inverse theorems, and applications
• Geometric and o-minimal Littlewood-Offord problems
• A non-uniform Littlewood-Offord inequality