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.

References in corpus (1)

• On a theorem of Schoen and Shkredov on sumsets of convex sets

Cited by in corpus (2)

• Some new inequalities in additive combinatorics
• Improved bounds on the set A(A+1)