Small Ball Probability, Inverse Theorems, and Applications

Abstract

Abstract. Let ξ be a real random variable with mean zero and variance one and A = {a1, . . . , an} be a multi-set in R d . The random sum SA := a1ξ1 + · · · + anξn where ξi are iid copies of ξ is of fundamental importance in probability and its applications.We discuss the small ball problem, the aim of which is to estimate the maximum probability that SA belongs to a ball with given small radius, following the discovery made by Littlewood-Offord and Erdős almost 70 years ago. We will mainly focus on recent developments that characterize the structure of those sets A where the small ball probability is relatively large. Applications of these results include full solutions or significant progresses of many open problems in different areas.

Arak’s inequalities for concentration functions and the Littlewood–Offord problem

“…Some of them have a non-empty intersection with the results of Nguyen, Tao and Vu [13,14,18,20] (see Theorem 2). Moreover, in [3], there are some structural results which would be apparently new in the Littlewood-Offord problem (see Theorems 3 and 4) and have no analogues in the literature.…”

“…In this paper, we apply Arak's results to the Littlewood-Offord problem which was intensively investigated in the last years. We compare the consequences of Arak's results with recent results of Nguyen, Tao and Vu [13], [14] and [18].Let X, X 1 , . .…”

“…The text on p. 10 of [3] is an analogue of descriptions of the Inverse Principles in the papers of Nguyen, Tao and Vu [13], [14] and [18]. A difference being that they restrict themselves to the classical Littlewood-Offord problem while discussing the arithmetic structure of the coefficients a 1 , .…”

“…This means that one deals with distributions of sums of nonidentically distributed random vectors of a special type. A further difference is that, in [13] and [14], the multivariate case is studied as well.…”

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“…Littlewood-Offord type results are commonly referred to as anti-concentration (or small-ball) inequalities. Anti-concentration results have been developed by many researchers through decades, and have recently found important applications in the theories of random matrices and random polynomials; see, for instance, [19] for a survey.…”