Constructing a Subsequence of (Exp(in))n∈N Converging towards Exp(iα) for a Given α∈R


For a given positive irrational and a real t ∈ [0,1), the explicit construction of a sequence of positive integers, such that the sequence of fractional parts of products converges towards t, is given. Moreover, a constructive and quantitative demonstration of the well known fact, that the ranges of the functions cos and sin are dense in the interval [-1,1], is presented. More precisely, for any α ∈ R, a sequence of positive integers is constructed explicitly in such a way that the estimate holds true for any jN. The technique used in the paper can give more general results, e.g. by replacing sine or cosine with continuous function f: RR having an irrational period.

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Lampret, V. (2015) Constructing a Subsequence of (Exp(in))n∈N Converging towards Exp(iα) for a Given α∈R. Open Access Library Journal, 2, 1-9. doi: 10.4236/oalib.1102135.

Conflicts of Interest

The authors declare no conflicts of interest.


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