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