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

Abstract

In this paper we study the behavior of concentration functions of weighted sums of independent random variables with respect to the arithmetic structure of coefficients. Recently, Tao and Vu formulated a so-called Inverse Principle in the Littlewood–Offord problem. We discuss the relations between this Inverse Principle and a similar principle formulated for sums of arbitrarily distributed independent random variables formulated by T. Arak in the 1980’s.