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March 23rd, 2011, 09:53 PM   #1
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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?
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March 23rd, 2011, 11:49 PM   #2
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Re: multiples of 2 pi

Hurwitz's theorem

Rewrite the inequality without modulus signs and then multiply everything by q.
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March 24th, 2011, 12:13 AM   #3
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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?
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March 24th, 2011, 04:32 AM   #4
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Re: multiples of 2 pi

[deleted]
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March 25th, 2011, 02:56 AM   #5
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Re: multiples of 2 pi

Deleted? Why did you do that?
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March 25th, 2011, 10:05 AM   #6
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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.
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March 25th, 2011, 10:16 AM   #7
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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.
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March 25th, 2011, 10:22 AM   #8
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Re: multiples of 2 pi

I restored the first post based on aswoods' recollection.
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March 25th, 2011, 01:11 PM   #9
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Re: multiples of 2 pi

Thanks CRG!
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