On the diameter and distribution of Sidon sets

Daniel Carter, Kevin O’Bryant, and I have uploaded to the arXiv our paper On the diameter and distribution of Sidon sets. This is a paper stemming from an interaction on my blog last year, regarding the use of AlphaEvolve to improve the then state of the art on the size of Sidon sets. We obtained a series of further refinements to the bound until progress stalled; after several months of no further progress, we decided to release what we have in this preprint. Recall that a Sidon set is a set of integers with the property that all the sums with and are distinct. Equivalently, all the differences , are distinct, except at where of course there is a large multiplicity. A basic question (Erdős problem #30) asks for the largest size of a Sidon set in . Several algebraic constructions of Sidon sets (by Singer, Bose, and Ruzsa) are known with and special choices of ; see this survey of O’Bryant for details. Erdős and Turán have conjectured that cannot exceed , with the bound also proposed…