A nonuniform Littlewood-Offord inequality for all norms

Abstract

Let ${\mathbf{v}}_i$ be vectors in ${\mathbb{R}}^d$ and $\{{\epsilon }_i\}$ be independent Rademacher random variables. Then the Littlewood-Offord problem entails finding the best upper bound for ${\sup }_{\mathbf{x}\in {\mathbb{R}}^d}\mathbb{P}(\sum {\epsilon }_i{\mathbf{v}}_i=\mathbf{x})$. Generalizing the uniform bounds of Littlewood-Offord, Erdős and Kleitman, a recent result of Dzindzalieta and Juškevičius provides a non-uniform bound that is optimal in its dependence on $\Vert \mathbf{x}{\Vert }_2$. 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 Juškevičius.

5 pages, corrected error in the proof of Theorem 1.4