  Let $X,X_1,...,X_n$ be independent identically distributed random variables.
In this paper we study the behavior of the concentration functions of the
weighted sums $\sum\limits_{k=1}^{n}a_k X_k$ with respect to the arithmetic
structure of coefficients $a_k$. Such concentration results recently became
important in connection with investigations about singular values of random
matrices. In this paper we formulate and prove some refinements of a result of
Vershynin (R. Vershynin, Invertibility of symmetric random matrices,
arXiv:1102.0300. (2011). Published in Random Structures and Algorithms, v. 44,
no. 2, 135--182 (2014)).
