  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 $\sum\limits_{k=1}^{n}X_k a_k$
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
multidimensional generalizations of the results Eliseeva and Zaitsev (2012).
They are also the refinements of the results of Friedland and Sodin (2007) and
Rudelson and Vershynin (2009).
