On lacunary sets for nonabelian groups

Abstract

Results concerning a class of lacunary sets are generalized from compact abelian to compact nonabelian groups. This class was introduced for compact abelian groups by Bozejko and Pytlik; it includes thep-Sidon sets of Edwards and Ross. A notion of test family is introduced and is used to give necessary conditions for a set to be lacunary. Using this, it is shown that y<B (2)

Central lacunary sets for Lie groups

“…This result is proved in the context of sets of type central A(p, q) and central V(p, q) previously discussed by the author [3]. (Note t h a t V ( l , p ' ) = p-Sidon.)…”

“…(Note t h a t V ( l , p ' ) = p-Sidon.) A few ancillary results are obtained from these sets, which enable one to fill some gaps left in [3], showing in particular t h a t these classes, too, are strictly larger than previously considered classes. (There are sets of type central V(s,r) which are not central A(2).…”

“…[8], [10]. In [3] it is shown that if G is semisimple, then G contains no infinite p-Sidon sets. It is shown [10] for general compact connected G that G contains an infinite central terms of use, available at https://www.cambridge.org/core/terms.…”

“…A subset of R of G is said to be of type central A(p, r) if there exists K\ G R so that for all sequences {a a ) a^R with a a = 0 off a finite set, [3] and of type central V(p,r) if there exists K2 € R so that for all such sequences <r€R P'…”

“…These sets have been discussed by the author in [2], [3]. (1.2) Let G be a compact connected Lie group of rank /, and let T = T l be a maximal torus for G. The Weyl group W is the finite group NG(T)/T, where NG{T) denotes the normalizer of T in G, and has an action on T given by w = gT: t -> wt = gtg~*.…”