Weighted p-Sidon sets

Abstract

A weighted generalization of a p-Sidon set, called an (a, p)-Sidon set, is introduced and studied for infinite, non-abelian, connected, compact groups G. The entire dual object G is shown never to be central

Central Interpolation Sets for Compact Groups and Hypergroups

“…This yields a new method of proving that every infinite subset of the dual of a compact, connected group admits an infinite central (a, 1)-Sidon set for any a < 1. Since a central (1/p, 1)-Sidon set is also central p-Sidon set [15] this approach also gives a new proof of the existence of central p-Sidon sets for p > 1, first established in non-abelian groups in [3].…”

“…The non-existence of infinite Sidon sets in many non-abelian groups is a consequence of the unboundedness of the degrees of the representations and motivated the introduction of weighted Sidon-type sets in [15] , where the effect of the degree is dampened.…”

“…Motivated by the fact that there are no infinite Sidon sets in the dual of any compact, simple, connected Lie group, the weaker notion of (central)(a, p)-Sidon sets was introduced in [15]. Our interest is in the case p = 1, and we extend the definition to (a, 1)-I 0 sets.…”

Central sidonicity for compact Lie groups

“…It is seen in [5] that if G is an infinite compact, connected group then G is never central (0, l)-Sidon, but there are examples where G is (-£, 1)-Sidon for any given e > 0. Also, every central (l4-£, l)-Sidon set for e > 0 is a set of representations of bounded degree; consequently our interest (when p = 1) is in the range 0 < a <: 1.…”

“…There are a number of equivalent characterizations of (central) (a,p)-Sidonicity (see [5]). For example, analogous to [6] …”