# Density in the unit circle

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• Feb 7th 2011, 07:04 AM
DrSteve
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.
• Feb 7th 2011, 09:20 AM
tonio
Quote:

Originally Posted by DrSteve
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
• Feb 7th 2011, 10:09 AM
Hartlw
Quote:

Originally Posted by DrSteve
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).
• Feb 7th 2011, 10:13 AM
Defunkt
Quote:

Originally Posted by Hartlw
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?
• Feb 7th 2011, 10:27 AM
Hartlw
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.
• Feb 7th 2011, 10:59 AM
FernandoRevilla
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
• Feb 7th 2011, 11:28 AM
Hartlw
FernandoRevilla
Can you give us a reference for "a well known theorem of Chaotic Dynamical Systems (Jacobi's theorem)," preferably on the internet?

Thanks
• Feb 7th 2011, 11:34 AM
FernandoRevilla
Quote:

Originally Posted by Hartlw
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.
• Feb 7th 2011, 11:47 AM
Hartlw
Quote:

Originally Posted by FernandoRevilla
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.
• Feb 7th 2011, 12:06 PM
FernandoRevilla
Quote:

Originally Posted by Hartlw
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.

Quote:

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

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

Quote:

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
• Feb 7th 2011, 12:12 PM
Hartlw
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).
• Feb 7th 2011, 12:33 PM
Hartlw
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?
• Feb 7th 2011, 01:44 PM
Defunkt
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.
• Feb 7th 2011, 02:48 PM
FernandoRevilla
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
• Feb 8th 2011, 01:03 AM
Hartlw
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
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