Good approximation and characterization of subgroups of R/Z

Biró, András and Deshouillers, J-M. and T. Sós, Vera (2001) Good approximation and characterization of subgroups of R/Z. STUDIA SCIENTIARUM MATHEMATICARUM HUNGARICA, 38 (1-4). pp. 97-113. ISSN 0081-6906

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Abstract

Let α be a real irrational number and A = (x n) be a sequence of positive integers. We call A a characterizing sequence of α or of the group ℤα mod 1 if lim ∥nβ∥ = 0 n∈A n→∞ if and only if β ∈ ℤα mod 1. In the present paper we prove the existence of such characterizing sequences, also for more general subgroups of ℝ/ℤ. In the special case ℤα mod 1 we give explicit construction of a characterizing sequence in terms of the continued fraction expansion of α. Further, we also prove some results concerning the growth and gap properties of such sequences. Finally, we formulate some open problems.

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