Lacunary Fourier series for compact quantum groups

TL;DR

This paper investigates the properties of Sidon sets, $\\Lambda(p)$-sets, and lacunarity in the context of compact quantum groups, establishing new characterizations and relations among these notions.

Contribution

It introduces new characterizations of Sidon sets, extends results to noncommutative $L^p$-spaces, and explores the structure of central Sidon sets in quantum groups.

Findings

Any Sidon set in a discrete group is a strong Sidon set.

Existence of $\Lambda(p)$-sets for orthogonal systems in noncommutative $L^p$-spaces.

Quantum groups with the same fusion rules have identical central Sidon sets.

Abstract

This paper is devoted to the study of Sidon sets, $\Lambda(p)$-sets and some related notions for compact quantum groups. We establish several different characterizations of Sidon sets, and in particular prove that any Sidon set in a discrete group is a strong Sidon set in the sense of Picardello. We give several relations between Sidon sets, $\Lambda(p)$-sets and lacunarities for $L^p$-Fourier multipliers, generalizing a previous work by Blendek and Michali\u{c}ek. We also prove the existence of $\Lambda(p)$-sets for orthogonal systems in noncommutative $L^p$-spaces, and deduce the corresponding properties for compact quantum groups. Central Sidon sets are also discussed, and it turns out that the compact quantum groups with the same fusion rules and the same dimension functions have identical central Sidon sets. Several examples are also included.