  Let $X,X_1,\ldots,X_n$ be independent identically distributed random
variables. In this paper we study the behavior of concentration functions of
weighted sums $\sum_{k=1}^{n}a_k X_k$ with respect to the arithmetic structure
of coefficients $a_k$ in the context of the Littlewood--Offord problem.
Concentration results of this type received renewed interest in connection with
distributions of singular values of random matrices. Recently, Tao and Vu
proposed an Inverse Principle in the Littlewood-Offord problem. We discuss the
relations between the Inverse Principle of Tao and Vu as well as that of Nguyen
and Vu and a similar principle formulated for sums of arbitrary independent
random variables in the work of Arak from the 1980's.
