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ISSN: 2333-9721
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Constructing a Subsequence of (Exp(in))n∈N Converging towards Exp(iα) for a Given α∈R

DOI: 10.4236/oalib.1102135, PP. 1-9

Subject Areas: Number Theory, Numerical Mathematics, Mathematical Analysis

Keywords: Convergence, Dense, Estimate, Exponential, Fractional Part, Integer Part, Irrational, Limit Point, Sequence

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Abstract

For a given positive irrationaland a real ∈ [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.

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.

References

[1]  Aliprantis, C.D. and Burkinshaw, O. (1999) Problems in Real Analysis—A Workbook with Solutions. Academic Press, Inc., San Diego.
[2]  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
[3]  Staib, J.H. and Demos, M.S. (1967) On the Limit Points of the Sequence {Sin n}Mathematics Magazine, 40, 210-213.
http://dx.doi.org/10.2307/2688681
[4]  Ogilvy, S.C. (1969) The Sequence {Sin n}Mathematics Magazine, 42, 94.
[5]  Luca, F. (1999) is dense in [-1,1]. Bulletin Mathématique de la Société des Sciences Mathématiques de RoumanieNouvelle Série, 42, 369-376.
[6]  Ahmadi, M.F. and Hedayatian, K. (2006) Limit Points of Trigonometric Sequences. Journal of Mathematical Extension, 1, 21-26.
[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
[8]  Wolfram, S. (1988-2008) Mathematica—Version 8.0. Wolfram Research, Inc., Champaign, IL.

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