  Let $X_1,\ldots,X_n$ be independent identically distributed random variables.
In this paper we study the behavior of concentration functions of weighted sums
$\sum_{k=1}^{n}X_ka_k $ with respect to the arithmetic structure of
coefficients~$a_k$ in the context of the Littlewood--Offord problem. In our
recent papers, we discussed the relations between the inverse principles stated
by Nguyen, Tao and Vu and similar principles formulated by Arak in his papers
from the 1980's. In this paper, we will derive some improved (more general and
more precise) consequences of Arak's inequalities applying our new bound in the
Littlewood--Offord problem. Moreover, we also obtain an improvement of the
estimates used in Rudelson and Vershynin's least common denominator method.
