Maximal additive growth of convex sets

Establish that every finite convex set A of n real numbers satisfies min(|A+A|,|A-A|) ≫_ε n^{2−ε} and E(A) ≪_ε n^{2+ε} for every ε>0.

Background

The paper studies additive structure through the notion of L3 control. Convex sets have essentially optimal control, so general bounds for controlled sets yield quantitative information about their sumsets, difference sets, and additive energy.

The stated conjecture, attributed in particular to Erdős, predicts that convex sets exhibit maximal possible additive growth. The paper improves the known bounds for additive energy and difference sets but does not establish the conjectured exponents for all three quantities.

References

It has long been conjectured (in particular by Erdős ) that convex sets should exhibit maximal growth, in that \min(|A+A|,|A-A|)\gg_\epsilon |A|{2-\epsilon} and E(A)\ll_\epsilon |A|{2+\epsilon}.

— Control and its applications in additive combinatorics  (2501.09470 - Bloom, 16 Jan 2025) in Section 1, subsection “Application 1: Convex sets”