Some new inequalities in additive combinatorics

Abstract

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.

39 pages

References in corpus (2)

• Some applications of W. Rudin's inequality to problems of combinatorial number theory
• Concentration points on two and three dimensional modular hyperbolas and applications

Cited by in corpus (1)

• Incomplete exponential sums over exponential functions