The apparent structure of dense Sidon sets

Sean Eberhard, Freddie Manners

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)
67 Downloads (Pure)


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 σn denotes the size of the largest Sidon subset of Z/nZ, this implies lim infn→∞ σn/n1/2 < 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.

Original languageEnglish
Article number#P1.33
JournalElectronic Journal of Combinatorics
Issue number1
Publication statusPublished - 24 Feb 2023

Bibliographical note

Funding Information:
∗SE has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 803711) and from the Royal Society. †FM is supported by a 2022 Sloan Research Fellowship.

Publisher Copyright:
© The authors.

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics
  • Applied Mathematics


Dive into the research topics of 'The apparent structure of dense Sidon sets'. Together they form a unique fingerprint.

Cite this