Central lacunary sets for Lie groups

Abstract

If G is a compact connected Lie group every infinite subset of G contains an infinite central A(p) set, for p < 2 + 2rankG/(dim G -rankG). A subset R of G is of type central A(2) if and only if the associated set of characters on the maximal torus is of type A (2). The dual of a compact connected semisimple Lie group contains infinite sets which are central p-Sidon for all p > 1. Every infinite subset of the dual of SU(2) contains such a set.

Central sidonicity for compact Lie groups

“…In [3], Dooley constructs in the dual of any compact, connected semisimple Lie group examples of infinite sets which are p-Sidon for all p > 1. By making the obvious modifications to his proof these examples can be seen to be central (a,p)-Sidon for all p > 1 and a < 2p-1.…”

“…The key idea of the proof (which we make precise below) is that "most" of H,, counted by multiplicity, lies outside 5,_i. This we are able to obtain from Lemma 2.3 and property (3). To be precise we have, if k > j,…”

“…This investigation is motivated in part by the fact that both Dooley's examples [3] of central p-Sidon sets and our examples from Theorem 2.1 correspond to Sidon sets in Z^"^. Because the representations a + p, a C G, belong to the fundamental Weyl chamber, the weights w(a + p) are distinct as w varies over W and a over E ( [7], ch.…”

“…-The second statement follows from the first since sets containing arbitrarily long arithmetic progressions are never Sidon sets in Z ( [8], p. 77). We follow the strategy of [3] Obviously there are many ways to choose a sequence {rij} containing arbitrarily long arithmetic progressions, and yet have Xk sufficiently small so that (*) bounded over all N and all a < 1. One choice, whose verification is routine, and is left for the reader, is to set n^k^ = A^l + z) for i = 0,1,..., 2^ -1, where A is sufficiently large.…”

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].…”