A non-abelian, non-Sidon, completely bounded $\Lambda(p)$ set

TL;DR

This paper constructs a specific subset within a non-abelian group that exhibits boundedness properties across all $p< finite$, yet does not fit traditional classifications like Leinert or Sidon sets, highlighting novel harmonic analysis phenomena.

Contribution

It provides the first example of a non-abelian, non-Sidon, completely bounded $\Lambda(p)$ set, expanding understanding of harmonic analysis on non-abelian groups.

Findings

Constructed a non-abelian group example with the desired properties.

Demonstrated the set is not a Leinert or weak Sidon set.

Showed the set is a completely bounded $\Lambda(p)$ set for all $p< finite$.

Abstract

The purpose of this note is to construct an example of a discrete non-abelian group $G$ and a subset $E$ of $G$, not contained in any abelian subgroup, that is a completely bounded $\Lambda (p)$ set for all $p<\infty ,$ but is neither a Leinert set nor a weak Sidon set.