Sidon sets are proportionally Sidon with small Sidon constants

TL;DR

This paper demonstrates that Sidon sets in torsion-free groups are proportionally highly independent and can be approximated by Sidon sets with arbitrarily small constants, strengthening their structural understanding.

Contribution

It introduces the concept of proportionally n-degree independence for Sidon sets and shows they can be approximated with Sidon constants close to one, a novel result.

Findings

Sidon sets in torsion-free groups are proportionally n-degree independent.

Sidon sets can be approximated with Sidon constants arbitrarily close to one.

Enhanced structural properties of Sidon sets are established.

Abstract

In his seminal work on Sidon sets, Pisier found an important characterization of Sidonicity: A set is Sidon if and only if it is proportionally quasi-independent. Later, it was shown that Sidon sets were proportionally `special' Sidon in several other ways. Here, we prove that Sidon sets in torsion-free groups are proportionally $n$-degree independent, a higher order of independence than quasi-independence, and we use this to prove that Sidon sets are proportionally Sidon with Sidon constants arbitrarily close to one, the minimum possible value.