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.
@article{10_37236_11191,
author = {Sean Eberhard and Freddie Manners},
title = {The apparent structure of dense {Sidon} sets},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {1},
doi = {10.37236/11191},
zbl = {1510.05018},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11191/}
}
TY - JOUR
AU - Sean Eberhard
AU - Freddie Manners
TI - The apparent structure of dense Sidon sets
JO - The electronic journal of combinatorics
PY - 2023
VL - 30
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/11191/
DO - 10.37236/11191
ID - 10_37236_11191
ER -
%0 Journal Article
%A Sean Eberhard
%A Freddie Manners
%T The apparent structure of dense Sidon sets
%J The electronic journal of combinatorics
%D 2023
%V 30
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/11191/
%R 10.37236/11191
%F 10_37236_11191
Sean Eberhard; Freddie Manners. The apparent structure of dense Sidon sets. The electronic journal of combinatorics, Tome 30 (2023) no. 1. doi: 10.37236/11191
