ACTA issues

Jamison sequences in countably infinite discrete Abelian groups

Vincent Devinck

Acta Sci. Math. (Szeged) 82:3-4(2016), 481-508

Abstract. We extend the definition of Jamison sequences in the context of topological abelian groups. We then study these sequences when the group is discrete and countably infinite. An arithmetical characterization of such sequences is obtained, extending the result of Badea and Grivaux [BadeaGrivaux2] about Jamison sequences of integers. In particular, we prove that the sequence consisting in all the elements of the group is a Jamison sequence. In the opposite, a sequence which generates a subgroup of infinite index in the group is never a Jamison sequence. We also generalize a result of Nikolskii by showing that the growth of the norms of a representation is influenced by the Haar measure of its unimodular point spectrum.

DOI: 10.14232/actasm-015-020-3

AMS Subject Classification (1991): 47A10, 37C85, 43A40, 28C10

Keyword(s): unimodular point spectrum, Jamison sequences, discrete abelian groups, characters and dual group, Haar measures

Received February 28, 2015, and in final form December 20, 2015. (Registered under 20/2015.)