TITLE: How often are irrational numbers well-approximated by rationals?
QUESTION [27 upvotes]: Suppose $x\in \mathbb{R}$ is irrational, with irrationality measure $\mu=\mu(x)$; this means that the inequality $|x-\frac{p}{q}|< q^{-\lambda}$ has infinitely many solutions in integers $p,q$ if and only if $\lambda < \mu$. A beautiful theorem of Roth asserts that algebraic numbers have irrationality measure $2$. For $\lambda<\mu$, let $\mathcal{Q}(x,\lambda) \subset \mathbb{N}$ be the (infinite) set of all $q$ occuring in solutions to the aforementioned inequality.
Question: For which pairs $(x,\lambda)$ does $\mathcal{Q}(x,\lambda)$ have positive relative density in the positive integers? For which pairs $(x,\lambda)$ does the cardinality of $\mathcal{Q}(x,\lambda) \cap [1,N]$ grow like a positive power of $N$?

REPLY [8 votes]: An answer to your second question may be found in an old paper of Erdos: Some results on diophantine approximation. Acta Arith. 5 1959 359–369 (1959). (MathSciNet)
He proves that for $1 < \lambda < 2$, for almost every $x$, the set $\mathcal{Q}(x,\lambda)$ grows like $n^\frac{1}{2-\lambda}$.  More precisely, Erdos proves that a one-sided version of your set $$\mathcal{Q}'(x,\lambda) = \big\{ q \geq 1 \ \big| \ \text{there exists } p, \ (q,p) = 1, \text{ such that } 0 < x-\frac pq < q^{-\lambda} \big\}$$ satisfies $$\lim_{n \to \infty} \frac{\big |\mathcal{Q}'(x,\lambda) \cap [1,n] \big| }{\sum_{q=1}^n q^{1-\lambda}} = \frac{12}{\pi^2}.$$ That the set $\mathcal{Q}(x,\lambda)$ grows like $n^\frac{1}{2-\lambda}$ is consistent with the fact that $\mathcal{Q}(x,1)$ is all of $\mathbb{N}$ while the numbers in $\mathcal{Q}(x,2)$ coming from continued fraction convergent denominators grow exponentially.