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> For which irrational numbers $\xi$ does there exist a constant $A$ such that $\left|\frac{p}{q}-\xi\right|<\frac{1}{Aq^2}$ (where $p/q$ is a rational number) has only finitely many solutions?
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Background
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I apologize if this is a terribly elementary question—my background is in analysis, not number theory. If there is some well-known source that answers this question, I would appreciate a link or citation.


As far as I can tell, there are definitely *some* irrational $\xi$ for which such an $A$ exists. For instance, if $\xi = \frac{1+\sqrt{5}}{2}$, then for any $A>\sqrt{5}$ the inequality $\left|\frac{p}{q}-\xi\right|<\frac{1}{Aq^2}$ has only finitely many solutions (this seems to be a very elementary result, which appears in a number of texts on elementary number theory). Additionally, if $\xi$ is an irrational root of a quadratic polynomial, then there is some $A$ for which $\left|\frac{p}{q}-\xi\right|<\frac{A}{q^2}$ has *no* solutions (see [the Wikipedia article on Liouville numbers](http://en.wikipedia.org/wiki/Liouville_number#Liouville_numbers_and_transcendence)). Thus for any irrational root of a quadratic polynomial, there is some constant $A$ of the kind desired.


According to MathWorld, there are some results in this direction: the article on [Hurwitz's Irrational Number Theorem](http://mathworld.wolfram.com/HurwitzsIrrationalNumberTheorem.html) suggests that the answer to this question is be related to the existence or nonexistence of a [Lagrange number](http://mathworld.wolfram.com/LagrangeNumber.html) associated to a particular irrational number. However, there are only countably many Lagrange numbers, and each Lagrange number is associated to only a countable number of irrationals, so it seems that the existence/nonexistence of a Lagrange number associated to a particular irrational is only a part of the answer.


Again, I apologize if this is a really basic question, and appreciate any help or advice.



Are there lower bounds on the quality of a rational approximation?
Here are some standard facts that you can find in many textbooks, e.g. in Cassels: An introduction to diophantine approximation.


1. There exists an $A>0$ such that $\left|\frac{p}{q}-\xi\right|<\frac{1}{Aq^2}$ has only finitely many rational solutions $\frac{p}{q}$ if and only if the continued fraction expansion of $\xi$ consists of bounded digits. In particular, quadratic irrationals have this property because they have a periodic continued fraction expansion.
2. There exists an $A<3$ such that $\left|\frac{p}{q}-\xi\right|<\frac{1}{Aq^2}$ has only finitely many rational solutions $\frac{p}{q}$ if and only if $\xi$ lies in an $\mathrm{SL}\_2(\mathbb{Z})$-orbit associated with some Markov number (see [here](http://en.wikipedia.org/wiki/Markov_number)). In particular, only countably many $\xi$'s belong to this case.
3. There are continuum many $\xi$'s such that $\left|\frac{p}{q}-\xi\right|\geq\frac{1}{3q^2}$ for any rational number $\frac{p}{q}$.


