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This paper explores the relationship between convexity and sum sets. Inparticular, we show that elementary number theoretical methods, principally theapplication of a squeezing principle, can be augmented with theElekes-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 areimprovements of recent results and takes advantage of computer algebra totackle some of the computations.

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