Arak inequalities for concentration functions and the Littlewood–Offord problem

Let $X,X_1,\ldots,X_n$ be independent identically distributed random variables. In this paper we study the behavior of the concentration functions of the weighted sums $\sum_{k=1}^{n}X_ka_k $ depending on the arithmetic structure of the coefficients $a_k$. The results obtained the last 10 years for the concentration functions of weighted sums play an important role in the study of singular numbers of random matrices. Recently, Tao and Vu proposed a so-called inverse principle for the Littlewood–Offord problem. We discuss the relations between this inverse principle and a similar principle for sums of arbitrarily distributed independent random variables formulated by Arak in the 1980s.