On the counting problem in inverse Littlewood–Offord theory

Let (Formula presented.) be independent and identically distributed Rademacher random variables taking values (Formula presented.) with probability (Formula presented.) each. Given an integer vector (Formula presented.), its concentration probability is the quantity (Formula presented.). The Littlewood–Offord problem asks for bounds on (Formula presented.) under various hypotheses on (Formula presented.), whereas the inverse Littlewood–Offord problem, posed by Tao and Vu, asks for a characterization of all vectors (Formula presented.) for which (Formula presented.) is large. In this paper, we study the associated counting problem: How many integer vectors (Formula presented.) belonging to a specified set have large (Formula presented.) ? 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, (Formula presented.) for some positive constant (Formula presented.)) 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 (Formula presented.), due to Cook; and (ii) dense row-regular (Formula presented.) -matrices, for which the previous best-known bound is (Formula presented.) for any constant (Formula presented.), due to Nguyen.