Affiliation:
1. 1 Please ask the editor of the journal.
2. 2 Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences Reáltanoda u. 13-15. 1053 Budapest
Abstract
Let be a real irrational number and A =(xn) be a sequence of positive integers. We call A a characterizing sequence of or of the group Z mod 1 if lim n 2A n !1 k k =0 if and only if 2 Z mod 1. In the present paper we prove the existence of such characterizing sequences, also for more general subgroups of R = Z . Inthespecialcase Z mod 1 we give explicit construction of a characterizing sequence in terms of the continued fraction expansion of. 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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