The Ubiquity of Sidon Sets That Are Not $I_0$

TL;DR

This paper demonstrates that in any infinite discrete abelian group, there exist Sidon sets that are not $I_0$, highlighting the widespread occurrence of such sets.

Contribution

It proves that every infinite discrete abelian group contains a Sidon set that is not $I_0$, expanding understanding of the structure of these sets.

Findings

Existence of non-$I_0$ Sidon sets in all infinite discrete abelian groups

Union of two $I_0$ sets may not be $I_0$

Every such group contains a Sidon set that is not $I_0$

Abstract

We prove that every infinite, discrete abelian group admits a pair of $I_0$ sets whose union is not $I_0$. In particular, this implies that every such group contains a Sidon set that is not $I_{0}$.