Let ${\epsilon }_1,\dots ,{\epsilon }_n$ be independent and identically distributed Rademacher random variables taking values $\pm 1$ with probability $1/2$ each. Given an integer vector $\mathit{a}=(a_1,\dots ,a_n)$, its concentration probability is the quantity $\rho (\mathit{a}):={sup}_{x\in \mathbb{Z}}Pr({\epsilon }_1{a}_1+\cdots +{\epsilon }_n{a}_n=x)$. The Littlewood–Offord problem asks for bounds on $\rho (\mathit{a})$ under various hypotheses on $\mathit{a}$, whereas the inverse Littlewood–Offord problem, posed by Tao and Vu, asks for a characterization of all vectors $\mathit{a}$ for which $\rho (\mathit{a})$ is large. In this paper, we study the associated counting problem: How many integer vectors $\mathit{a}$ belonging to a specified set have large $\rho (\mathit{a})$? The motivation for our study is that in typical applications, the inverse Littlewood–Offord theorems are only used to obtain such counting estimates. Using a more direct approach, we obtain significantly better bounds for this problem than those obtained using the inverse Littlewood–Offord theorems of Tao and Vu and of Nguyen and Vu. Moreover, we develop a framework for deriving upper bounds on the probability of singularity of random discrete matrices that utilizes our counting result. To illustrate the methods, we present the first ‘exponential-type’ (that is, $\exp (-c{n}^c)$ for some positive constant $c$) upper bounds on the singularity probability for the following two models: (i) adjacency matrices of dense signed random regular digraphs, for which the previous best-known bound is $O(n^{-1/4})$, due to Cook; and (ii) dense row-regular $\{0,1\}$-matrices, for which the previous best-known bound is $O_C(n^{-C})$ for any constant $C>0$, due to Nguyen.

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关于逆 Littlewood-Offord 理论中的计数问题

让 ${\epsilon }_1,\dots ,{\epsilon }_n$ 是取值的独立同分布的 Rademacher 随机变量 $\pm 1$ 有概率 $1/2$每个。给定一个整数向量$\mathit{一种}=({\mathrm{一种}}_1,\dots ,{\mathrm{一种}}_n)$，其集中概率为数量 $\rho (\mathit{一种})：={超}_{X\in \mathbb{Z}}压力({\epsilon }_1{\mathrm{一种}}_1+\cdots +{\epsilon }_n{\mathrm{一种}}_n=X)$. Littlewood-Offord 问题要求在$\rho (\mathit{一种})$ 在各种假设下 $\mathit{一种}$，而由 Tao 和 Vu 提出的逆Littlewood-Offord 问题要求对所有向量进行表征$\mathit{一种}$ 为此 $\rho (\mathit{一种})$很大。在本文中，我们研究了相关的计数问题：有多少整数向量$\mathit{一种}$ 属于一个指定的集合有大 $\rho (\mathit{一种})$? 我们研究的动机是在典型应用中，逆 Littlewood-Offord 定理仅用于获得此类计数估计。使用更直接的方法，我们获得了比使用 Tao 和 Vu 以及 Nguyen 和 Vu 的逆 Littlewood-Offord 定理获得的边界更好的边界。此外，我们开发了一个框架，用于利用我们的计数结果推导出随机离散矩阵奇异性概率的上限。为了说明这些方法，我们展示了第一个“指数类型”（即，$\mathrm{经验值}(-C{n}^C)$ 对于一些正常数 $C$) 以下两个模型的奇点概率上限： (i) 稠密有符号随机正则有向图的邻接矩阵，其中先前最著名的界限是 $哦(n^{-1/4})$，由于库克；(ii) 密集排正则$\{0,1\}$- 矩阵，其中先前最著名的界限是 ${哦}_C(n^{-C})$ 对于任何常数 $C>0$，由于阮。