Small ball estimates for quasi-norms

TL;DR

This paper provides new small ball probability bounds and Littlewood-Offord estimates for random vectors in finite-dimensional spaces with quasi-norms, extending previous results to more general settings.

Contribution

It introduces two types of small ball estimates for quasi-norms, generalizing earlier work by Friedland, Sodin, Rudelson, and Vershynin.

Findings

Bounds for small ball probabilities under density smoothness

Littlewood-Offord type estimates for quasi-norms

Extension of previous results to broader quasi-norm settings

Abstract

This note contains two types of small ball estimates for random vectors in finite dimensional spaces equipped with a quasi-norm. In the first part, we obtain bounds for the small ball probability of random vectors under some smoothness assumptions on their density function. In the second part, we obtain Littlewood-Offord type estimates for quasi-norms. This generalizes a result which was previously obtained by Friedland and Sodin and by Rudelson and Vershynin.