Constructing a Subsequence of (Exp(i&em&n&/em&))&sub&&em&n&/em&∈N&/sub& Converging towards Exp(i&em&α&/em&) for a Given &em&α&/em&∈R

摘要

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 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.