TITLE:
Constructing a Subsequence of (Exp(in))n∈N Converging towards Exp(iα) for a Given α∈R
AUTHORS:
Vito Lampret
KEYWORDS:
Convergence, Dense, Estimate, Exponential, Fractional Part, Integer Part, Irrational, Limit Point, Sequence
JOURNAL NAME:
Open Access Library Journal,
Vol.2 No.12,
December
31,
2015
ABSTRACT:
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.