Improved sumset bounds for convex sets

Improve the lower bound for the sumset of a finite convex set A beyond |A+A| ≫_ε |A|^{30/19−ε}, or establish the conjectured near-quadratic growth |A+A| ≫_ε |A|^{2−ε}.

Background

The paper obtains improved bounds for the additive energy and difference set of convex sets, but its sumset estimate does not improve the previously known exponent 30/19.

This unresolved limitation concerns a concrete quantitative problem: strengthening the lower bound for |A+A| for ordinary convex sets, with the broader maximal-growth conjecture predicting an exponent arbitrarily close to 2.

References

For |A+A| we have been unable to improve on the bound |A+A|\gg_\epsilon |A|{30/19-\epsilon} of Rudnev and Stevens .

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