Near optimal three-fold additive energy bound for points on convex curves

Abstract

Let $X\subset\mathbb{R}$ be finite and let $\gamma(t)=(t,f(t))$, where $f$ is strictly convex. We show that \[ J_3(\gamma(X)) =\#\{(x_1,\ldots,x_6)\in X^6:\sum_{i=1}^3\gamma(x_i)=\sum_{i=4}^6\gamma(x_i)\} \ll_{\varepsilon}|X|^{3+\varepsilon}. \] As applications, we prove that $|A-A|\gg_{\varepsilon}|A|^{5/3-\varepsilon}$ and $|A+A|\gg_{\varepsilon}|A|^{8/5-\varepsilon}$ for any convex sequence $A\subset \mathbb{R}$.