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

For a given positive irrationaland 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 j ∈ N. The technique used in the paper can give more general results, e.g. by replacing sine or cosine with continuous function f: R→R having an irrational period.