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Math Help - Density in the unit circle

  1. #1
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    Density in the unit circle

    Here's an interesting Complex Analysis question. It's probably not too difficult, but I'm just not seeing the solution for some reason:

    Show that \{ e^{ki}|k\in \mathbb{Z},k\geq 0\} is dense in the unit circle.


    So I think this reduces to the following question:

    Given \epsilon >0 and \theta \in \mathbb{R}, show that there are integers n and k with k\geq 0 such that

    |e^{i(\theta - k+2n\pi)}-1|<\epsilon.

    A hint how to proceed would be nice.
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  2. #2
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    Quote Originally Posted by DrSteve View Post
    Here's an interesting Complex Analysis question. It's probably not too difficult, but I'm just not seeing the solution for some reason:

    Show that \{ e^{ki}|k\in \mathbb{Z},k\geq 0\} is dense in the unit circle.


    So I think this reduces to the following question:

    Given \epsilon >0 and \theta \in \mathbb{R}, show that there are integers n and k with k\geq 0 such that

    |e^{i(\theta - k+2n\pi)}-1|<\epsilon.

    A hint how to proceed would be nice.

    Look at theorem 2.1, much more general than what you want, here:

    http://www.math.msu.edu/~shapiro/pub...s/nonmsble.pdf

    There they talk of [0,1) modulo 1 instead of the circle group, and in the proof where it

    says "k+1 distinct elements of G" it must be "k+1 distinct elements of H"

    Tonio
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    Quote Originally Posted by DrSteve View Post
    Here's an interesting Complex Analysis question. It's probably not too difficult, but I'm just not seeing the solution for some reason:

    Show that \{ e^{ki}|k\in \mathbb{Z},k\geq 0\} is dense in the unit circle.


    So I think this reduces to the following question:

    Given \epsilon >0 and \theta \in \mathbb{R}, show that there are integers n and k with k\geq 0 such that

    |e^{i(\theta - k+2n\pi)}-1|<\epsilon.

    A hint how to proceed would be nice.
    Rudin's (and Shilov's) definition of dense:

    If X is a metric space and E is a subset of X,
    E is dense in X if every point of X is a limit point of E or a point of E(or both).

    So by that definition { e^{ki} can't be dense in the unit circle.

    EDIT: However, if X is the unit circle, the question boils down to the behavior of cosn+isinn on the unit circle as n-> infinity (what points are included).
    Last edited by Hartlw; February 7th 2011 at 11:20 AM.
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  4. #4
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    Quote Originally Posted by Hartlw View Post
    Rudin's (and Shilov's) definition of dense:

    If X is a metric space and E is a subset of X,
    E is dense in X if every point of X is a limit point of E or a point of E(or both).

    So by that definition { e^{ki} can't be dense in the unit circle.
    Why not?
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    OK, I was a little terse. If X is the metric space all points in the unit circle and E is the set e^ki, then there are points in X which aren't a limit point or point of E. But see Edit to my previous post.
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  6. #6
    MHF Contributor FernandoRevilla's Avatar
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    We can apply a well known theorem of Chaotic Dynamical Systems (Jacobi's theorem) :

    Theorem

    Let \lambda\in\mathbb{R} and consider:

    T_{\lambda}:S^1\rightarrow S^1,\quad T_{\lambda}(\theta)=\theta +2\pi\lambda

    Then, each orbit of T_{\lambda} is dense in S^1 if \lambda is irrational

    Particular case

    In our case we choose \lambda=1/2\pi (irrational) , so:

    O^+=\{T^k(0):k\in\mathbb{N}\}=\{e^{ki}:k\in\mathbb  {N}\}

    is dense in S^1 .


    Fernando Revilla
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    FernandoRevilla
    Can you give us a reference for "a well known theorem of Chaotic Dynamical Systems (Jacobi's theorem)," preferably on the internet?

    Thanks
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    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by Hartlw View Post
    FernandoRevilla
    Can you give us a reference for "a well known theorem of Chaotic Dynamical Systems (Jacobi's theorem)," preferably on the internet?

    It is Theorem 3.13 of Chaotic Dynamical Systems by Robert L. Devaney. I have no references on the Internet.


    Fernando Revilla


    P.S. If someone is interested, I can provide the proof.
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    Quote Originally Posted by FernandoRevilla View Post
    We can apply a well known theorem of Chaotic Dynamical Systems (Jacobi's theorem) :

    Theorem

    Let \lambda\in\mathbb{R} and consider:

    T_{\lambda}:S^1\rightarrow S^1,\quad T_{\lambda}(\theta)=\theta +2\pi\lambda

    Then, each orbit of T_{\lambda} is dense in S^1 if \lambda is irrational

    Particular case

    In our case we choose \lambda=1/2\pi (irrational) , so:

    O^+=\{T^k(0):k\in\mathbb{N}\}=\{e^{ki}:k\in\mathbb  {N}\}

    is dense in S^1 .


    Fernando Revilla
    Why does this apply to post #1. Can you correlate the terms?

    Is T for example the assignment of e^ki to k? on the unit circle S1 (roughly)? I have the feeling the theorem is just a rewording of the original post, in which case the original post hasn't been answered.

    EDIT I suspect your theorem says cosn+isinn covers all points on the unit circle for n=0 to infinity, in which case I would appreciate seeing a proof of it because I find it an interesting question in its own right.
    Last edited by Hartlw; February 7th 2011 at 12:59 PM.
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  10. #10
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by Hartlw View Post
    Is T for example the assignment of e^ki to k? on the unit circle S1 (roughly)?

    We denote a point in S^1 by its angle \theta in the standard manner.

    I have the feeling the theorem is just a rewording of the original post,

    No, it isn't a rewording. It is a generalization.


    in which case the original post hasn't been answered.

    Proving the Jacobi's theorem the problem has been "more than answered". If we don't want to generalize we only need to substitute \lambda by 1/2\pi in the mentioned proof.


    Fernando Revilla
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    To state that a generalization of the original theorem has been proven somewhere is not really in the spirit of the original question.

    How about a proof of orignal post, perhaps more preciseley reworded?

    EDIT: But recognizing the orig post as the special case of a more general theorem is a valid reply, and maybe I'm being a little contentious, though I would still like to see a proof of original post (sooner or later cosn+isinn will hit me and then I can prove original post).
    Last edited by Hartlw; February 7th 2011 at 01:23 PM.
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  12. #12
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    You just need to show cosn takes on all values between 0 and 1 for some n. If you plot cosn versus n in Excel you get a cosine curve which "looks" continuous. Is that a legit new thread?
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  13. #13
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    Hartlw, it is impossible that \{e^{ik} | k \in \mathbb{N} \} takes on all values of the unit circle S^1, simply because the cardinality of the former set is stricly smaller than that of the second ( \aleph_0 and \aleph, respectively).
    The same applies to "cosn takes all values between 0 and 1" because, again, \{cos(n) | n \in \mathbb{N} \} is countable, while [0,1] is not.
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  14. #14
    MHF Contributor FernandoRevilla's Avatar
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    Theorem

    Let \lambda\in\mathbb{R} and consider:

    T_{\lambda}:S^1\rightarrow S^1,\quad T_{\lambda}(\theta)=\theta +2\pi\lambda

    Then, each orbit of T_{\lambda} is dense in S^1 if \lambda is irrational

    Proof

    Let \theta\in S^1 . The points on the orbit of \theta are distinct for if T_{\lambda}(\theta)^n=T_{\lambda}^m(\theta) we would have (m-n)\lambda\in \mathbb{Z}, so that n=m . Any infinite set of points on the circle must have a limit point. Thus, given any \epsilon >0 there must be integers n and m for which |T_{\lambda}^n(\theta)-T_{\lambda}^m(\theta)|<\epsilon . Let k=n-m . Then |T_{\lambda}^k(\theta)-\theta|<\epsilon .

    Now T_{\lambda} preserves lengths in S^1 . Consequently, T_{\lambda}^k maps the arc connecting \theta to T_{\lambda}^k(\theta) to the arc connecting T_{\lambda}^k(\theta) and T_{\lambda}^{2k}(\theta) which has length less than \epsilon . In particular it follows that the points \theta,T_{\lambda}^k(\theta),T_{\lambda}^{2k}(\the  ta),\ldots partition S^1 into arcs of length less than \epsilon . Since \epsilon was arbitrary, this completes the proof.


    Fernando Revilla
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    I sense FernandoRevilla is correct though I can't follow it (I am not qualified to judge). Defunkt states, in effect, a countably infinite set of points in a real interval doesn't include all the points. That is not the question at hand, though I admit I was a little sloppy in some previous phrasing about "all points."

    From an Engineering perspective:

    e^ik is a point on the unit circle at an angle of k radians.

    If I proceed around the unit circle in steps of k radians do I ever hit the same point again, ie, after m steps do I move a multiple of 2pi? Or, does k exist such that m=2pik? Is pi rational? No.

    So I have a countably infinite number of points on the circle. Are they dense?

    Somehow this has to get to the equivalent question, are the rational points on [0,1] dense? Yes, because every real point of the interval is a limit point of the rational numbers on the interval.

    An infinite number of points on the circle must indeed have a limit point, but is every point hit as per above a limit point, in terms I could understand? In other words, given any point on the circle, are there points constructed as per above on the circle within an arc length epsilon?

    EDIT: So I have the question, given any point on the unit circle, how many times do I have to go around to get to within a distance epsilon? Getting close.

    OK: For any angle a and any desired degree of accuracy (couched in terms of epsilon),
    m/n<=a<=(m+1)/n
    m<=na<=(m+1)
    k=m
    Last edited by Hartlw; February 8th 2011 at 04:01 AM.
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