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?

## Background

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). 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 suggests that the answer to this question is be related to the existence or nonexistence of a Lagrange number 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.