Quantcast

Got Homework?

Connect with other students for help. It's a free community.

  • across
    MIT Grad Student
    Online now
  • laura*
    Helped 1,000 students
    Online now
  • Hero
    College Math Guru
    Online now

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

across

I am having trouble understanding the theorem below, so I will post it here as a reminder to myself to review it as often as I can. Also, I may get lucky enough to have someone clarify the questions that I will most likely post here as I read through it time and time again.

  • 2 years ago
  • 2 years ago

  • This Question is Closed
  1. across
    Best Response
    You've already chosen the best response.
    Medals 3

    Dirichlet's Approximation Theorem "If \(\alpha\) is a real number and \(n\) is a positive integer, then there exist integers \(a\) and \(b\) with \(1\leq a\leq n\) such that \(|a\alpha-b|<1/n\)." Proof. Consider the \(n+1\) numbers \(0,\{\alpha\},\{2\alpha\}\), ..., \(\{n\alpha\}\). These \(n+1\) numbers are the fractional parts of the numbers \(j\alpha\), \(j=0,1\), ..., \(n\), so that \(0\leq\{j\alpha\}<1\) for \(j=0,1\), ..., \(n\). Each of these \(n+1\) numbers lies in one of the \(n\) disjoint intervals \(0\leq x<1/n\), \(1\leq x<2/n\), ..., \((j-1)\leq x<j/n\), ..., \((n-1)\leq x<1\). Because there are \(n+1\) number under consideration, but only \(n\) intervals, the pigeonhole principle tells us that at least two of these numbers lie in the same interval. Because each of these intervals has length \(1/n\) and does not include its right endpoint, we know that the distance between two numbers that lie in the same interval is less than \(1/n\). It follows that there exist integers \(j\) and \(k\) with \(0\leq j < k\leq n\) such that \(|\{k\alpha\}-\{j\alpha\}|<1/n\). Now let \(a=k-j\) and \(b=[k\alpha]-[j\alpha]\). Because \(0\leq j < k\leq n\), we see that \(1\leq a\leq n\). Moreover,\[|a\alpha-b|=|(k-j)\alpha-([k\alpha]-[j\alpha])|\]\[=|(k\alpha-[k\alpha])-(j\alpha-[j\alpha])|\]\[=|\{k\alpha\}-\{j\alpha\}|<1/n.\]Consequently, we have found integers \(a\) and \(b\) with \(1\leq a\leq n\) and \(|a\alpha-b|<1/n\), as desired. \(\blacksquare\)

    • 2 years ago
  2. asnaseer
    Best Response
    You've already chosen the best response.
    Medals 0

    Your question prompted me to search this topic and try and learn about it. During my google quest, I came across the original German paper on this topic and thought it might interest you as I /think/ you are a German speaker. http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/anzeige?band=07-abh/1837&seite:int=00000286

    • 2 years ago
  3. No-data
    Best Response
    You've already chosen the best response.
    Medals 0

    Sabes algo de eso jose?

    • 2 years ago
  4. victorarana
    Best Response
    You've already chosen the best response.
    Medals 0

    si

    • 2 years ago
    • Attachments:

See more questions >>>

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.