From: israel@math.ubc.ca (Robert Israel) Subject: Re: Density of U_n=n*sin(n) ? Date: 26 Apr 1999 21:37:05 GMT Newsgroups: sci.math In article , Bruno LANGLOIS wrote: >Is the sequence U_n=n*sin(n) dense in IR ? Tim Chow asked exactly the same question in sci.math.research on December 8. Here was my reply: -------------------------------------- In article <74c545\$s10@schubert.mit.edu>, Tim Chow writes: |> Is the set {n sin n} dense in the reals? I'm pretty sure this is unknown. Let's take the easier question, does n sin n have values arbitrarily close to 0? In order for lim inf |n sin n| = 0, you need the existence of good rational approximations for pi: for every c > 0 there must be positive integers m and n with |pi - n/m| < c/m^2. This is equivalent to the continued fraction for pi having unbounded elements. It is not known whether this is the case, although almost every real does have unbounded elements. For the stronger condition that { n sin n } is dense, you need the following: for every c > 0 and every real t there are positive integers m and n such that |pi - n/m - t/m^2| < c/m^2. This would in particular imply that for every integer k >= 2, infinitely many elements of the continued fraction are k or k+1. --------------------------------------- I might add the corollary: for { n sin(a n) } to be dense, infinitely many elements of the continued fraction of pi/a must be k or k+1. So (since we know the continued fraction for e) { n sin(pi/e n) } is not dense. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2