  Let $X,X_1,...,X_n$ be independent identically distributed random variables.
The paper deals with the question about the behavior of the concentration
function of the random variable $\sum_{k=1}^{n}a_k X_k$ according to the
arithmetic structure of coefficients $a_k$. 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 of Friedland and Sodin (2007) and Rudelson and
Vershynin (2009).
