March 23rd, 2011, 09:53 PM  #1 
Newbie Joined: Mar 2011 Posts: 12 Thanks: 0  multiples of 2 pi
How can I show that multiples of 2? are dense mod 1? Can a multiple be found which is arbitrarily close to some integer?

March 23rd, 2011, 11:49 PM  #2 
Senior Member Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3  Re: multiples of 2 pi 
March 24th, 2011, 12:13 AM  #3 
Senior Member Joined: Nov 2010 Posts: 502 Thanks: 0  Re: multiples of 2 pi
One can do this with any irrational number  take k +1 multiples mod 1 and note that two must be within 1/k of eachother, and that none repeat (because it's irrational and you're modding out by a rational number). But then this small difference can be repeated and so we can get arbitrarily dense multiples of any irrational number (i.e. for any k). This was a bare sketch because it's just a pigeonhole argument (or a rethinking of Dirichlet's box theorem). Does that make sense?

March 24th, 2011, 04:32 AM  #4 
Newbie Joined: Mar 2011 Posts: 12 Thanks: 0  Re: multiples of 2 pi
[deleted]

March 25th, 2011, 02:56 AM  #5 
Senior Member Joined: Nov 2010 Posts: 502 Thanks: 0  Re: multiples of 2 pi
Deleted? Why did you do that?

March 25th, 2011, 10:05 AM  #6 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,204 Thanks: 511 Math Focus: Calculus/ODEs  Re: multiples of 2 pi
When you delete your posts containing the problem (particularly the first post), you render the entire topic useless to anyone else. Please refrain from this practice, as it strikes many here as selfish.

March 25th, 2011, 10:16 AM  #7 
Senior Member Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3  Re: multiples of 2 pi
He was asking how to show that multiples of 2? were dense mod 1, and/or whether a multiple could be found which was arbitrarily close to the closest integer.

March 25th, 2011, 10:22 AM  #8 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: multiples of 2 pi
I restored the first post based on aswoods' recollection.

March 25th, 2011, 01:11 PM  #9 
Senior Member Joined: Nov 2010 Posts: 502 Thanks: 0  Re: multiples of 2 pi
Thanks CRG!


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