%0 Journal Article
%T Constructing a Subsequence of (Exp(i<em>n</em>))<sub><em>n</em>¡ÊN</sub> Converging towards Exp(i<em>¦Á</em>) for a Given <em>¦Á</em>¡ÊR
%A Vito Lampret
%J Open Access Library Journal
%V 2
%N 12
%P 1-9
%@ 2333-9721
%D 2015
%I Open Access Library
%R 10.4236/oalib.1102135
%X For a given positive irrational<img src="http://www.oalib.com:80/image/DEF_PAPERATTACHED_PATH/20151231100613_274.bmp" alt="" />and a real <strong><em>t&nbsp;</em></strong>¡Ê [<strong>0</strong>,<strong>1</strong>), the explicit
construction of a sequence <img src="http://www.oalib.com:80/image/DEF_PAPERATTACHED_PATH/20151231100732_440.bmp" alt="" />of positive
integers, such that the sequence of fractional parts of products&nbsp;<img src="http://www.oalib.com:80/image/DEF_PAPERATTACHED_PATH/20151231100800_318.bmp" alt="" />converges towards <i>t</i>, 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 [<strong>-1</strong>,<strong>1</strong>], is presented. More precisely,
for any <em><strong>¦Á&nbsp;</strong></em>¡Ê R, a sequence <img src="http://www.oalib.com:80/image/DEF_PAPERATTACHED_PATH/20151231100941_243.bmp" alt="" />&nbsp;of positive integers is constructed explicitly
in such a way that the estimate <img src="http://www.oalib.com:80/image/DEF_PAPERATTACHED_PATH/20151231100959_521.bmp" alt="" />&nbsp;holds true for any <strong><em>j</em></strong> ¡Ê <strong>N</strong>. The technique used in
the paper can give more general results, e.g. by replacing sine or cosine with
continuous function <strong><em>f</em>: R</strong><strong>¡ú</strong><strong>R</strong> having an irrational period.
%K Convergence
%K Dense
%K Estimate
%K Exponential
%K Fractional Part
%K Integer Part
%K Irrational
%K Limit Point
%K Sequence
%U http://www.oalib.com/paper/3152632