Let @math be i.i.d. Rademacher random variables taking values @math with probability @math each. Given an integer vector @math , its concentration probability is the quantity @math . The Littlewood-Offord problem asks for bounds on @math under various hypotheses on @math , whereas the inverse Littlewood-Offord problem, posed by Tao and Vu, asks for a characterization of all vectors @math for which @math is large. In this paper, we study the associated counting problem: How many integer vectors @math belonging to a specified set have large @math ? 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., @math for some positive constant @math ) 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 @math due to Cook; and (ii) dense row-regular @math -matrices, for which the previous best known bound is @math for any constant @math due to Nguyen.
This paper makes two contributions to the areas of anti-concentration and non-asymptotic random matrix theory. First, we study the counting problem in inverse Littlewood-Offord theory for general random variables: for random variables @math which are i.i.d. copies of a random variable @math (satisfying some mild hypotheses), how many integer vectors @math in a prescribed box have large @math ? Building on recent work of Ferber, Jain, Luh, and Samotij (who treated the case when @math is a Rademacher random variable), we provide significantly better bounds for this problem than those obtained using the inverse Littlewood-Offord theorems of Tao and Vu, and Nguyen and Vu. Next, we study the non-asymptotic behavior of the least singular value @math of a random @math square matrix @math with i.i.d. entries, which are only assumed to be centered with variance @math . As an application of our counting theorem, and utilizing and developing a recent work by the author, we show that for all @math , @math , where @math is an absolute constant, thereby providing a considerably simpler proof of an approximate version of an essentially optimal result due to Rebrova and Tikhomirov. An approximate Spielman-Teng theorem for the least singular value @math of a random @math square matrix @math is a statement of the following form: there exist constants @math such that for all @math , @math . The goal of this paper is to develop a simple and novel framework for proving such results for discrete random matrices. As an application, we prove an approximate Spielman-Teng theorem for @math -valued matrices, each of whose rows is an independent vector with exactly @math zero components. This improves on previous work of Nguyen and Vu, and is the first such result in a truly combinatorial' setting. Let @math denote a random symmetric @math matrix whose upper diagonal entries are independent and identically distributed Bernoulli random variables (which take values @math and @math with probability @math each). It is widely conjectured that @math is singular with probability at most @math . On the other hand, the best known upper bound on the singularity probability of @math , due to Vershynin (2011), is @math , for some unspecified small constant @math . This improves on a polynomial singularity bound due to Costello, Tao, and Vu (2005), and a bound of Nguyen (2011) showing that the singularity probability decays faster than any polynomial. In this paper, improving on all previous results, we show that the probability of singularity of @math is at most @math for all sufficiently large @math . The proof utilizes and extends a novel combinatorial approach to discrete random matrix theory, which has been recently introduced by the authors together with Luh and Samotij.