Strong characterizing sequences for subgroups of compact groups

Abstract

In [A. Biró, V.T. Sós, Strong characterizing sequences in simultaneous Diophantine approximation, J. Number Theory 99 (2003) 405–414] we proved that if Γ is a subgroup of the torus R / Z generated by finitely many independent irrationals, then there is an infinite subset A ⊆ Z which characterizes Γ in the sense that for γ ∈ R / Z we have ∑ a ∈ A ‖ a γ ‖ < ∞ if and only if γ ∈ Γ . Here we consider a general compact metrizable Abelian group G instead of R / Z , and we characterize its finitely generated free subgroups Γ by subsets A ⊆ G * , where G * is the Pontriagin dual of G. For this case we prove stronger forms of the analogue of the theorem of the above mentioned work, and we find necessary and sufficient conditions for a kind of strengthening of this statement to be true.