The apparent structure of dense Sidon sets

TL;DR

This paper explores the structure of dense Sidon sets in abelian groups, linking them to projective planes, classifying those from desarguesian planes, and conjecturing all dense sets originate from such geometric constructions.

Contribution

It classifies dense Sidon sets from desarguesian planes, explores examples from nondesarguesian planes, and conjectures a geometric origin for all dense Sidon sets.

Findings

All known dense Sidon sets from desarguesian planes are classified.

Many dense Sidon sets from nondesarguesian planes are identified.

A conjecture is proposed that all dense Sidon sets originate from projective plane constructions.

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, but there are many further examples arising from nondesarguesian planes. We conjecture that all dense Sidon sets arise in this manner. 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.