Sharpening an Estimate of the Size of the Sumset of a Convex Set

A finite set $A=\{a_1<\dotsb<a_n\}\subset\mathbb R$ is said to be convex if the sequence $(a_i-a_{i-1})_{i=2}^n$ is strictly increasing. Using an estimate of the additive energy of convex sets, one can estimate the size of the sumset as $|A+A|\gtrsim|A|^{102/65}$, which slightly sharpens Shkredov's latest result $|A+A|\gtrsim|A|^{58/37}$.