On the counting problem in inverse Littlewood--Offord theory

Let $\epsilon_1, \dotsc, \epsilon_n$ be i.i.d. Rademacher random variables taking values $\pm 1$ with probability $1/2$ each. Given an integer vector $\boldsymbol{a} = (a_1, \dotsc, a_n)$, its concentration probability is the quantity $\rho(\boldsymbol{a}):=\sup_{x\in \mathbb{Z}}\Pr(\epsilon_1 a_1+\dots+\epsilon_n a_n = x)$. The Littlewood-Offord problem asks for bounds on $\rho(\boldsymbol{a})$ under various hypotheses on $\boldsymbol{a}$, whereas the inverse Littlewood-Offord problem, posed by Tao and Vu, asks for a characterization of all vectors $\boldsymbol{a}$ for which $\rho(\boldsymbol{a})$ is large. In this paper, we study the associated counting problem: How many integer vectors $\boldsymbol{a}$ belonging to a specified set have large $\rho(\boldsymbol{a})$? 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' (i.e., $\exp(-n^c)$ 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.