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 <fo2ibdrhu7my@forum.swarthmore.edu>,
Bruno LANGLOIS <blang@club-internet.fr> 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