The Relationship between $\epsilon$-Kronecker and Sidon Sets

TL;DR

This paper investigates the relationship between $ ext{epsilon}$-Kronecker sets and Sidon sets in discrete abelian groups, proving that $ ext{epsilon}$-Kronecker sets with $ ext{epsilon}<2$ are indeed Sidon sets using Pisier's net characterization.

Contribution

The paper proves that $ ext{epsilon}$-Kronecker sets with $ ext{epsilon}<2$ are Sidon sets, establishing a significant link between these two classes of sets in harmonic analysis.

Findings

$ ext{epsilon}$-Kronecker sets with $ ext{epsilon}<2$ are Sidon sets.

Uses Pisier net characterization to establish the result.

Shows that $ ext{epsilon}$-Kronecker sets share arithmetic properties with Sidon sets.

Abstract

A subset $E$ of a discrete abelian group is called $\epsilon $-Kronecker if all $E$-functions of modulus one can be approximated to within $\epsilon$ by characters. $E$ is called a Sidon set if all bounded $E$-functions can be interpolated by the Fourier transform of measures on the dual group. As $\epsilon$-Kronecker sets with $\epsilon <2$ possess the same arithmetic properties as Sidon sets, it is natural to ask if they are Sidon. We use the Pisier net characterization of Sidonicity to prove this is true.