The Apparent Structure of Dense Sidon Sets

Abstract

The correspondence between perfect difference sets and transitive projective planes is well-known. We observe that all known dense (i.e., close to square-root size) Sidon subsets of abelian groups come from projective planes through a similar construction. We classify the Sidon sets arising in this manner from desarguesian planes and find essentially no new examples. There are many further examples arising from nondesarguesian planes. We conjecture that all dense Sidon sets arise from finite projective planes in this way. If true, this implies that all abelian groups of most orders do not have dense Sidon subsets. In particular if $\sigma_n$ denotes the size of the largest Sidon subset of $\mathbb{Z}/n\mathbb{Z}$, this implies $\liminf_{n \to \infty} \sigma_n / n^{1/2} < 1$. We also give a brief bestiary of somewhat smaller Sidon sets with a variety of algebraic origins, and for some of them provide an overarching pattern.