Some remarks on the asymmetric sum--product phenomenon

Abstract

Using some new observations connected to higher energies, we obtain quantitative lower bounds on $\max \{|AB|,|A+C|\}$ and $\max \{|(A+\alpha )B|,|A+C|\}$, $\alpha \ne 0$ in the regime when the sizes of finite subsets $A,B,C$ of a field differ significantly.

28 pages

References in corpus (6)

• Some new inequalities in additive combinatorics
• New results on sum-product type growth over fields
• Energies and structure of additive sets
• A short proof of a near-optimal cardinality estimate for the product of a sum set
• On distinct cross-ratios and related growth problems
• Expanders with superquadratic growth

Cited by in corpus (3)

• On growth rate in $S{L}_2({\mathbf{F}}_p)$, the affine group and sum-product type implications
• On the size of the set $AA+A$
• On popular sums and differences of sets with small products