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.
Cite this paper
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, e2135. doi: http://dx.doi.org/10.4236/oalib.1102135.
Radulescu,
T.-L.T., Radulescu, V.D. and Andreescu, T. (2009) Problems in Real Analysis—Advanced
Calculus on the Real Axis. Springer, Dodrecht, Heidelberg, New York. http://dx.doi.org/10.1007/978-0-387-77379-7
Zheng, S. and Cheng, J.C. (1999)
Density of the Images of Integers under Continuous Functions with Irrational
Periods. Mathematics Magazine, 72,
402-404. http://dx.doi.org/10.2307/2690800