A nonuniform Littlewood-Offord inequality for all norms

Abstract

Let $\mathbf{v}_i$ be vectors in $\mathbb{R}^d$ and $\{\varepsilon_i\}$ be independent Rademacher random variables. Then the Littlewood-Offord problem entails finding the best upper bound for $\sup_{\mathbf{x} \in \mathbb{R}^d} \mathbb{P}(\sum \varepsilon_i \mathbf{v}_i = \mathbf{x})$. Generalizing the uniform bounds of Littlewood-Offord, Erd\H{o}s and Kleitman, a recent result of Dzindzalieta and Ju\v{s}kevi\v{c}ius provides a non-uniform bound that is optimal in its dependence on $\|\mathbf{x}\|_2$. In this short note, we provide a simple alternative proof of their result. Furthermore, our proof demonstrates that the bound applies to any norm on $\mathbb{R}^d$, not just the $\ell_2$ norm. This resolves a conjecture of Dzindzalieta and Ju\v{s}kevi\v{c}ius.