Convexity, Squeezing, and the Elekes-Szabó Theorem

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-Szabó Theorem in order to give new information. Namely, if we let A ⊂ R, we prove that there exist a, a^′ ∈ A such that ∣_∣∣∣∣ (aA + 1)⁽²⁾ (a^′ A + 1)⁽²⁾ (aA + 1)⁽²⁾ (a^′ A + 1) ∣^≳|A|31/12. We are also able to prove that max{|A + A − A|, |A² + A² − A² |, |A³ + A³ − A³ |} ≳ |A|^19/12. Both of these bounds are improvements of recent results and takes advantage of computer algebra to tackle some of the computations.