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 Asubset of or equal toZ which characterizes Γ in the sense that for γset membership, variantR/Z we have ∑aset membership, variantAdouble vertical baraγdouble vertical bar<∞ if and only if γset membership, variantΓ. 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 Asubset of or equal toG*, 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.
