Some new inequalities in additive combinatorics

TL;DR

This paper introduces new inequalities in additive combinatorics, providing bounds on additive energy and sumsets for subsets like multiplicative subgroups and convex sets, advancing understanding of their additive structure.

Contribution

It presents novel inequalities involving set intersections and applies them to derive bounds on additive energy and sumsets for special subsets in abelian groups.

Findings

New bounds for additive energy of multiplicative subgroups

Sharp lower bounds for sumset cardinality of subgroups and subprogressions

Inequalities connecting additive energy and higher moments of convolutions

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.