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September 17th, 2008, 10:25 AM   #1
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Showing something is dense in ....

x,y elements of R s.t x=y if |x-y|=k where k is an integer.
=> topological circle with circumference 1

Let B be an irrational real s.t. 0<B<1...

look at the set B,2B,... and show that it is dense in 0<= x <= 1
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September 17th, 2008, 06:11 PM   #2
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Re: Showing something is dense in ....

this is what I have thus far.

given B is an element of R\Q, WTS that {kB mod 1 for positive integers k} is dense in [0,1]

so we need for all x in [0,1] and all episilon>0, there exists a k s.t. |(kB mod 1) -x| < epsilon.

so if we let epsilon = 1/n, where n is a natural number, then we want to find show that the distance between
kBmod1 and x is less than 1/n.

So if [0,1] is split up in n intervals, how do I show that there must exist some k so kB mod 1 is in the same interval as x?
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September 17th, 2008, 09:36 PM   #3
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Re: Showing something is dense in ....

I think the key is that it can't avoid any interval unless it loops, and if it loops it's rational. Perhaps when it's not 2AM I'll be able to make that slightly more coherent.
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September 18th, 2008, 05:06 PM   #4
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Re: Showing something is dense in ....

It has come to my attention that this is a well-known problem, better known as the Equidistribution theorem.

If this rings a bell to anyone or somehow helps, I'm still working on this.

I know that if there are some positive integers k,l s.t lB mod1 - kBmod1 < 1/n...
and l-k >0, then I can say that I can use multiples of p = l-k to span the unit interval.

However, if l-k < 0 then I would need some negative multiple of p, but I am only working with positive coefficients of B,

which is my problem. Any takers? Essentially I am able to move along one direction of the interval (or circle), but I need to know how to span in the other direction, well how to articulate/justify that.
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September 19th, 2008, 04:51 AM   #5
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Re: Showing something is dense in ....

Looking the problem over once more, I don't think that there is an easy proof, unless you have access to some strong theorems already. This is essentially ergodic theory... hard stuff.
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September 19th, 2008, 07:12 AM   #6
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Re: Showing something is dense in ....

Hmm. I mean, I think I'm supposed to use pigeonhole logic to solve this. If I know that an interval has at least two points in it...

I have seen the technical proofs for the problem, but I think there is a somewhat qualitative rationale that can be applied

by breaking the interval into say n pieces and showing that if you can cover one interval then you can cover all.
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September 19th, 2008, 07:44 AM   #7
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Re: Showing something is dense in ....

Yes, this density problem is well-known and makes use of the dirichlet principle. Assume x is irrational on R modulo 1 (that is, the circle with circumference 1). Then among the numbers k*x mod 1, where k is an integer ranging from 0 to say n>0, where n is large enough, we can always find two numbers which are as close to each other as desired (distance being measure by absolute value). Assume these two numbers are a*x mod 1 and b*x mod 1. Then |a-b| is very small (and n will depend on how close we want them to be). But then using multiples of (a-b)*x mod 1 you can approach any element of (0,1) with the precision (a-b)*x mod 1, which helps you solve your question.
I haven't taken the time to formalize the above properly but a formal proof can be trivially written from the above arguments. Hope this helps !
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September 24th, 2008, 07:28 AM   #8
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Re: Showing something is dense in ....

Yes. This is what I ended up doing. The main trouble was considering the case where (a-b) < 0. We want to show that
there are positive integer multiples of the irrational that span the unit interval. So when this difference is negative, we end up needing to use negative multiples. Essentially need to formalize how to span backwards for the unit interval.
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September 24th, 2008, 07:30 AM   #9
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Re: Showing something is dense in ....

An application of this proposition: show that given a string of digits (for instance 12323636748489), you can find a positive integer n such that the decimal expension of 2^n begins with the chosen string.
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