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.
Categories: math.CO math.NT