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 (10)

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• Additive energy of cyclic matrix groups and character sums with matrix exponential functions
• Improving cardinality estimation of sums of sets with convexity
• On tripling constant of multiplicative subgroups
• Sumsets of the distance set in ${\mathbb{F}}_q^d$
• On the exponential large sieve inequality for sparse sequences modulo primes