Convexity, Squeezing, and the Elekes-Szab\'{o} Theorem

TL;DR

This paper combines elementary number theory and the Elekes-Szabó Theorem to derive new bounds on sum sets and related expressions, improving previous results with computational assistance.

Contribution

It introduces novel bounds on sum set expressions using a combination of convexity, squeezing principles, and the Elekes-Szabó Theorem, enhanced by computer algebra methods.

Findings

Established lower bounds for complex sum set expressions.

Improved upon recent bounds for sum set sizes.

Utilized computer algebra to handle computational complexity.

Abstract

This paper explores the relationship between convexity and sum sets. In particular, we show that elementary number theoretical methods, principally the application of a squeezing principle, can be augmented with the Elekes-Szab\'{o} Theorem in order to give new information. Namely, if we let $A \subset \mathbb R$, we prove that there exist $a,a' \in A$ such that \[\left | \frac{(aA+1)^{(2)}(a'A+1)^{(2)}}{(aA+1)^{(2)}(a'A+1)} \right | \gtrsim |A|^{31/12}.\] We are also able to prove that \[ \max \{|A+A-A|, |A^2+A^2-A^2|, |A^3 + A^3 - A^3|\} \gtrsim |A|^{19/12}.\] Both of these bounds are improvements of recent results and takes advantage of computer algebra to tackle some of the computations.