论文标题
对所有规范的不均匀的利特伍德 - 局部不平等不平等
A nonuniform Littlewood-Offord inequality for all norms
论文作者
论文摘要
令$ \ mathbf {v} _i $为$ \ mathbb {r}^d $和$ \ {\ varepsilon_i \} $中的向量是独立的rademacher随机变量。然后，Littlewood-offord问题需要找到最佳的上限，以$ \ sup _ {\ MathBf {x} \ in \ Mathbb {r}^d} \ Mathbb {p}（\ sum \ sum \ varepsilon_i \ Mathbf {v} Dzindzalieta和Juškevičius的最新结果，概括了Littlewood-offord，Erdős和Kleitman的统一边界，提供了一种非均匀的界限，其依赖于$ \ | \ | \ Mathbf {x}} \ | _2 $是最佳的。在此简短说明中，我们提供了简单的替代证明其结果。此外，我们的证明表明，该界限适用于$ \ Mathbb {r}^d $上的任何规范，而不仅仅是$ \ ell_2 $ norm。这解决了Dzindzalieta和Juškevičius的猜想。
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, Erdős and Kleitman, a recent result of Dzindzalieta and Juškevičius 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 Juškevičius.
