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

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Cited by in corpus (20)

• Congruences involving product of intervals and sets with small multiplicative doubling modulo a prime and applications
• Energies and structure of additive sets
• Some remarks on the asymmetric sum--product phenomenon
• On sums of Szemerédi--Trotter sets
• New results on sum-products in R
• Double Character Sums over Subgroups and Intervals
• Intersections of multiplicative subgroups and Heilbronn's exponential sum
• An introduction to higher energies and sumsets
• Almost all primes have a multiple of small Hamming weight
• Incomplete exponential sums over exponential functions
• Any small multiplicative sugroup is not a sumset
• On exponential sums over multiplicative subgroups of medium size
• Rational exponential sums over the divisor function
• Additive energy of cyclic matrix groups and character sums with matrix exponential functions
• Exponential sums with sparse polynomials over finite fields
• On the exponential large sieve inequality for sparse sequences modulo primes
• Sumsets of the distance set in ${\mathbb{F}}_q^d$
• On tripling constant of multiplicative subgroups
• A question of Bukh on sums of dilates
• Improving cardinality estimation of sums of sets with convexity