On Sum Sets and Convex Functions

Abstract

In this paper we prove new bounds for sums of convex or concave functions. Specifically, we prove that for all $A,B \subseteq \mathbb R$ finite sets, and for all $f,g$ convex or concave functions, we have $$|A + B|^{38}|f(A) + g(B)|^{38} \gtrsim |A|^{49}|B|^{49}.$$

This result can be used to obtain bounds on a number of two-variable expanders of interest, as well as to the asymmetric sum-product problem. We also adjust our technique to prove the three-variable expansion result $$|AB+A|\gtrsim |A|^{\frac{3}{2} +\frac{3}{170}}\,.$$

Our methods follow a series of recent developments in the sum-product literature, presenting a unified picture. Of particular interest is an adaptation of a regularisation technique of Xue, originating in a paper of Rudnev, Shakan, and Shkredov, that enables us to find positive proportion subsets with certain desirable properties.