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

Abstract

Let $X\subset\mathbb{R}$ be finite and let $γ(t)=(t,f(t))$, where $f$ is strictly convex. We show that \[ J_3(γ(X)) =\#\{(x_1,\ldots,x_6)\in X^6:\sum_{i=1}^3γ(x_i)=\sum_{i=4}^6γ(x_i)\} \ll_ε|X|^{3+ε}. \] When specialized to the parabola, our result implies near-optimal estimates for the number of solutions to the diameter-free quadratic Vinogradov system. As a second application, we settle a conjecture from Krishnapur-Kurlberg-Wigman and Bombieri-Bourgain concerning lattice points on dilates of the unit circle. As a third application, we prove that $|A-A|\gg_ε|A|^{5/3-ε}$ and $|A+A|\gg_ε|A|^{8/5-ε}$ for any finite convex sequence $A\subset \mathbb{R}$.