A bound on the multiplicative energy of a sum set and extremal sum-product problems

TL;DR

This paper investigates extremal cases in sum-product problems by establishing a new upper bound on the multiplicative energy of sum and difference sets, shedding light on the tightness of existing bounds.

Contribution

It introduces a novel upper bound on the multiplicative energy of sum and difference sets, advancing understanding of extremal sum-product phenomena.

Findings

Provides extremal results for sum-product estimates

Establishes a new upper bound on multiplicative energy

Offers insights into the tightness of sum-product bounds

Abstract

In recent years some near-optimal estimates have been established for certain sum-product type estimates. This paper gives some first extremal results which provide information about when these bounds may or may not be tight. The main tool is a new result which provides a nontrivial upper bound on the multiplicative energy of a sum set or difference set.