On sum sets of 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 also prove the three-variable expansion result \[ |AB+A|\gtrsim |A|^{\frac32 +\frac3{170}}\,. \] 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, that enables us to find positive proportion subsets with certain desirable properties.

16 pages plus appendix

References in corpus (2)

• On sums of Szemerédi--Trotter sets
• Asymmetric estimates and the sum-product problems