Good approximation and characterization of subgroups of R/Z

Let a be a real irrational number and A = (x(n)) be a sequence of positive integers. We call A a characterizing sequence of a: or of the group Z alpha mod 1 if lim(n --> infinity n epsilonA) \\n beta\\ = 0 if and only if beta epsilon Z alpha mod 1. In the present paper we prove the existence of such characterizing sequence s, also for more general subgroups of R/Z. In the special case Z alpha mod 1 we give explicit construction of a characterizing sequence in terms of th e continued fraction expansion of alpha. Further, we also prove some result s concerning the growth and gap properties of such sequences. Finally, we f ormulate some open problems.