Welcome to MathOverflow

Question

10

2

More precisely, what real numbers $r$ have the following property: for any $\epsilon > 0$ there exist infinitely many pairs $(p, q)$ of integers such that

$$\left| \frac{p}{q} - r \right| < \frac{\epsilon}{q^2}.$$

I think that this is impossible if $r$ is a quadratic irrational. On the other hand, it's certainly possible for any number with irrationality measure strictly greater than $2$.

What I really want to know is if the real numbers which don't have the property above have measure zero. If that's true, it would answer the last part of this math.SE question.

edited Oct 25 2010 at 20:18

Douglas Zare
13.5k●3●36●59

asked Oct 24 2010 at 14:05

Qiaochu Yuan
38.8k●5●117●306

Felipe Voloch · Accepted Answer · 2010-10-24 14:30:48Z UTC

13

This is a well-studied question in diophantine approximation. You can look up Markov or Lagrange spectrum for a "description" of the numbers for which you cannot take $\epsilon$ arbitrarily small. For the answer to your last question, look up Khinchin's theorem (the answer is no, they have full measure).

answered Oct 24 2010 at 14:30

Felipe Voloch
16.7k●2●36●61

Thanks, Felipe! I can't seem to find a statement of Khinchin's theorem (I assume you're not referring to the theorem about Khinchin's constant) online. Do you have a reference? – Qiaochu Yuan Oct 24 2010 at 14:47

2

eom.springer.de/d/d032580.htm – Felipe Voloch Oct 24 2010 at 14:56

If I read it correctly, the answer is yes, almost all real numbers can be approximated pretty well by rationals. – Douglas Zare Oct 25 2010 at 20:14

1

For intuition, large coefficients in the simple continued fraction expansion produce good rational approximations. There is a limiting distribution for the coefficients, the Gauss-Kuzmin distribution, which is supported on all positive integers, so you expect that with some coarse independence that you get infinitely many coefficients greater than any fixed size except on a set of measure $0$. – Douglas Zare Oct 25 2010 at 20:32

@Douglas: Now I am confused, too many double negations. Most numbers can be well approximated. – Felipe Voloch Oct 25 2010 at 20:40

Hany · Answer 2 · 2010-10-26 19:35:50Z UTC

1

Hardy and Wright devoted a chapter (chapter 9) to these questions. One interesting theorem related to your question is theorem 196.

edited Oct 26 2010 at 19:35

answered Oct 25 2010 at 18:53

Hany
122●9

What numbers can be approximated “pretty well” by rationals?

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

2 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.