What numbers can be approximated "pretty well" by rationals? - MathOverflow

What numbers can be approximated "pretty well" by rationals?

2010-10-24T14:05:58Z

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.

Answer by Felipe Voloch for What numbers can be approximated "pretty well" by rationals?

2010-10-24T14:30:48Z

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).

Answer by Hany for What numbers can be approximated "pretty well" by rationals?

2010-10-25T18:53:36Z

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