On the counting problem in inverse Littlewood–Offord theory

Abstract

Let ε1,…,εn be independent and identically distributed Rademacher random variables taking values ±1 with probability 1/2 each. Given an integer vector a=(a1,…,an), its concentration probability is the quantity ρfalse(bold-italicafalse):=trueprefixsupx∈ZPrfalse(ε1a1+⋯+εnan=xfalse). The Littlewood–Offord problem asks for bounds on ρ(a) under various hypotheses on bold-italica, whereas the inverse Littlewood–Offord problem, posed by Tao and Vu, asks for a characterization of all vectors bold-italica for which ρ(a) is large. In this paper, we study the associated counting problem: How many integer vectors bold-italica belonging to a specified set have large ρ(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(−cnc) 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 OCfalse(n−Cfalse) for any constant C>0, due to Nguyen.

Resilience of the rank of random matrices

“…The main challenge is to prove (1.1), as it involves a union bound over all possible a ∈ R n . In order to overcome this difficulty, we use some recently developed machinery introduced in [4]. Roughly speaking, we embed the problem into a sufficiently large finite field F p .…”

“…Roughly speaking, we embed the problem into a sufficiently large finite field F p . Then, as there are finitely many options for a ∈ F p in the left kernel of M, we can use a counting argument from [4] to bound the probability of encountering each possible kernel vector a according to the corresponding value of ρ(a).…”

“…[8], [11], [12], [14] and [16]). The novelty of our argument is that we utilize the methods in [4] to obtain the bound (1.1). Most of the previously used arguments yield exponential or polynomial probabilities which would only tolerate a sublinear number of modifications to the matrix.…”

“…Lastly, we mention that the method in [4] has already been successfully applied to a variety of combinatorial problems in random matrix theory [2,3,6,7,10].…”

“…The remainder of this paper is organized as follows. In Section 2 we provide the necessary background to state the counting lemma from [4]. In Section 2.3 we provide a convenient interface to apply the counting lemma.…”