Improvement of the convex-set sumset bound

Improve the known lower bound |A+A|≫ε |A|^(30/19−ε) for finite convex sets A of real numbers, with the broader goal of establishing the conjectured near-quadratic sumset growth.

Background

The paper proves stronger estimates for the additive energy and difference set of convex sets, but it records no improvement for the sumset beyond the result of Rudnev and Stevens.

This unresolved gap is significant because the conjectured maximal-growth statement requires a sumset lower bound close to |A|², whereas the currently cited exponent 30/19 is substantially smaller.

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”