inequalities - How often are irrational numbers well-approximated by rationals?

$mathcal{Q}(x,lambda)$ has positive relative density if and only if $lambdale 1$.
This follows from Weyl's Theorem on Uniform Distribution. (There is a nice concise proof in Cassels' "Diophantine Approximation".)

Weyl's Theorem: Let $Isubset mathbb{R}$ be an interval of length $epsilon le 1$. Let $S_N(I)$ be the set of all integers $q$ in the interval $[1,N]$ such that for some integer $p$, it holds that $xq-pin I$. Then

$$frac{Card(S_N(I))}{N} to epsilon
text{ as } Ntoinfty.$$

Here's a proof-sketch, using Weyl's Theorem, that if $lambda > 1$ then $mathcal{Q}(x,lambda)$ has relative density zero:

Fix $epsilon > 0$, and take $I$ (in Weyl's Theorem) to be the interval $(-epsilon,epsilon)$. Suppose $lambda>1$. Let $qin mathcal{Q}(x,lambda)$; so for some $pin mathbb{Z}$, $$|xq-p| < q^{1-lambda}.$$

There is an integer $M$, depending only on $epsilon$, such that $|xq-p| < epsilon$ whenever $p$ and $q$ satisfy the above inequality and $qge M$. Therefore $$mathcal{Q}(x,lambda)cap [M,N]subset S_N(I).$$ It follows from Weyl's Theorem that the relative density of $mathcal{Q}(x,lambda)$ does not exceed $2epsilon$. Since $epsilon$ is arbitrary, the relative density of $mathcal{Q}(x,lambda)$ must be zero.

This can be proved in a more elementary but laborious way using the "Ostrowski Number System", which is explained in the Rockett and Szusz book on continued fractions.