  In the paper we find new inequalities involving the intersections $A\cap
(A-x)$ of shifts of some subset $A$ from an abelian group. We apply the
inequalities to obtain new upper bounds for the additive energy of
multiplicative subgroups and convex sets and also a series another results on
the connection of the additive energy and so--called higher moments of
convolutions. Besides we prove new theorems on multiplicative subgroups
concerning lower bounds for its doubling constants, sharp lower bound for the
cardinality of sumset of a multiplicative subgroup and its subprogression and
another results.
