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

Let $X\subset \mathbb{R}$ be finite and let $\gamma (t)=(t,f(t))$, where $f$ is strictly convex. We show that $$J_3(\gamma (X))=\mathrm{\#}\{(x_1,\dots ,x_6)\in X^6:\sum _{i=1}^3\gamma (x_i)=\sum _{i=4}^6\gamma (x_i)\}{\ll }_{\epsilon }\mathrm{\mid }X{\mathrm{\mid }}^{3+\epsilon }\mathrm{.}$$ As applications, we prove that $\mathrm{\mid }A-A\mathrm{\mid }{\gg }_{\epsilon }\mathrm{\mid }A{\mathrm{\mid }}^{5\mathrm{/}3-\epsilon }$ and $\mathrm{\mid }A+A\mathrm{\mid }{\gg }_{\epsilon }\mathrm{\mid }A{\mathrm{\mid }}^{8\mathrm{/}5-\epsilon }$ for any convex sequence $A\subset \mathbb{R}$.