Multivariate estimates for the concentration functions of weighted sums of independent identically distributed random variables

This article is a multidimensional generalization of the results Eliseeva and Zaitsev (2012). Let $X,X_1,\ldots,X_n$ be independent identically distributed random variables. The paper deals with the question about the behavior of the concentration function of the random variable $\sum_{k=1}^na_kX_k$ according to the arithmetic structure of vectors $a_k$. Recently the interest to this question has increased significantly due to the study of distributions of eigenvalues of random matrices. In this paper we formulate and prove some refinements of the results Friedland and Sodin (2007) and Rudelson and Vershynin (2009).