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I'm having trouble building intuition for whether or not a set is dense in another set. I've read this reference, which was helpful, but I'm having a few outstanding questions.

The examples provided in the link are obvious, and they don't really give me a good sense of how I would go about writing proofs or dis-proofs of denseness.

For example, I was posed the following two problems that, just looking at the provided reference, I don't have a clue how to tackle:

Are the following sets dense in $[0, 1]$?

  • $\{\sin n \,| \,n \in \Bbb{N}\}$
  • $\{frac(\pi^n) \,| \,n \in \Bbb{N}\}$

Where $frac$ is the fractional part of a number.

Are there any (better) references out there that can help me better understand not only what denseness is, but also how I should start proofs of this type?

Community
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anonymouse
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  • One good way to think about dense sets is via limits or neighborhoods. A Set $A$ is dense in $[0,1]$ if any interval containing $x \in [0,1]$ contains also a point in $A$. We can then formulate this using epsilon language. Via sequences, one should be able to have a sequence in $A$ converging to any point in $[0,1]$. – DaveNine Mar 08 '16 at 22:37
  • To add to my comment since I can't figure out how to edit on my phone, think about how $\mathbb{Q}$ is dense in $\mathbb{R}$. No matter how small of an interval you take surrounding a real number, you can be sure that your interval is going to contain a rational number as well. – DaveNine Mar 08 '16 at 22:39
  • But that's my confusion. It's really easy to think about the set $\Bbb{Q}$, but I don't really know what the sets in my original post look like. Yeah, they're a bunch of non-terminating real numbers, and maybe I'll even concede that it makes sense that it should be dense in $[0, 1]$, but I don't know how to think about these specific sets in terms of neighborhoods. – anonymouse Mar 08 '16 at 22:43
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    If it is not yet clear, the issue does not seem to be your understanding of what dense means, but rather it seems to be a case of your examples being particularly difficult (without the right tools from other areas of mathematics). For example, someone can have a very good understanding of what an irrational number is and still not be able to prove that ${\pi}^5 - 8{\pi}^3$ is irrational. – Dave L. Renfro Mar 08 '16 at 22:49
  • ${\sin(n) : n\in\mathbb N}$ might be dense in $[-1,1]$, but not in $[0,1]$. – Friedrich Philipp Mar 08 '16 at 22:54
  • Take a point $x \in [0,1]$. For any $\epsilon>0$, can you find $n\in \mathbb{N}$ so that $|x-\sin(n)|<\epsilon$? Think about how $\sin{n}$ would distribute itself around a unit circle. Shouldn't we be able to find some interval where this can't be? – DaveNine Mar 08 '16 at 23:05
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    I think I'm missing something big here. Wouldn't a set being dense in $[-1, 1]$ imply that the set is dense in any of the subsets, namely $[0, 1]$? Also, to DaveNine's comment, $\sin n$ feels like it should cover the whole unit circle eventually, right? Two points can never overlap because integers can never differ by any multiple of $2\pi$, so it feels like we can find $n$. Where am I going wrong here? – anonymouse Mar 09 '16 at 00:07
  • You question about $\text{frac}(π^n)$ could be difficult : see here or there. – Watson Mar 27 '16 at 10:48

1 Answers1

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For sets of this type is well to remember the Equidistribution Theorem : https://en.m.wikipedia.org/wiki/Equidistribution_theorem . Morever you should note that the functions sine and $ frac $ are continuos and surjective in $[0,1)$ . In fact , if $f:X\rightarrow Y$ is continuos and surjective ($X, Y$ are topological spaces) and $T\subset X$ is dense, then $f (T)$ is dense in $ Y $.

Vincenzo Zaccaro
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  • Wait, but the equidistribution theorem doesn't help with the sequence $\pi^n$. And in what sense is frac continuous? You should probably be a little more explicit about what map you are using to represent $\sin$. – Erick Wong Mar 08 '16 at 22:50
  • And how can one get from ${n\pi\mod 1: n\in\mathbb N}$ (dense in $[0,1]$) to proving that ${n\mod\pi : n\in\mathbb N}$ is dense in $[0,\pi]$? – Friedrich Philipp Mar 08 '16 at 23:06
  • @FriedrichPhilipp There's no easy way between those two sets directly, but you can get from "${n/\pi\mod 1: n\in\mathbb N}$ is dense in $[0,1]$" to "${n\mod\pi : n\in\mathbb N}$ is dense in $[0,\pi]$" by composing with the homeomorphism $f(x)=\pi x$ from $[0,1]$ to $[0,\pi]$, which maps dense sets to dense sets. – Mario Carneiro May 01 '16 at 04:12