The Sidon error term

Abstract: A Sidon set is a set ${\mathcal A}$ of integers that has no nontrivial solutions to $a+b=c+d$. It has been known since 1941 (Erd\H{o}s and Tur\'an) that if ${\mathcal A}$ is a finite Sidon set, then $\text{diam}({\mathcal A}) \ge k^2 - 2k^{3/2} + O(k)$, and since 1939 (Singer) that the $k^2$ term cannot be improved. Only in the last 5 years has the error term $-2k^{3/2}$ been sharpened (Balogh, F\"uredi, and Roy, then O'Bryant, then Carter, Hunter, O'Bryant). In this talk, I will relay the latest improvements and applications, and the use of AI (AlphaEvolve) in their discovery. Joint work with D.~Carter, B.~Georgiev, Z.~Hunter, J.~G.~Serrano, T.~Tao, and A.~Zs.~Wagner.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)