Convexity and a sum-product type estimate

Abstract: In this paper we further study the relationship between convexity and additive growth, building on the work of Schoen and Shkredov (\cite{SS}) to get some improvements to earlier results of Elekes, Nathanson and Ruzsa (\cite{ENR}). In particular, we show that for any finite set $A\subset{\mathbb{R}}$ and any strictly convex or concave function $f$, [|A+f(A)|\gg{\frac{|A|^{24/19}}{(\log|A|)^{2/19}}}] and [\max{|A-A|,\ |f(A)+f(A)|}\gg{\frac{|A|^{14/11}}{(\log|A|)^{2/11}}}.] For the latter of these inequalities, we go on to consider the consequences for a sum-product type problem.