The set $A\_\tau$ of irrational numbers $x$ which are $\tau$-approximable, i.e., that satisfy the estimate
$$\left|x - \frac{p}{q}\right| \leq \frac{1}{q^\tau}$$
for infinitely many rationals $p/q$, has Hausdorff dimension $2/\tau$. This is the so called Jarník-Besicovitch theorem. What can be said about the $2/\tau$-Hausdorff outer measure of $A\_\tau$? Is it always positive or infinite? If one replaces $1/q^\tau$ with $C/q^\tau$ for some $C > 0$, how does it depend on $C$?


EDIT: Some remarks:


1) The set $A\_\tau$ is 1-periodic in the sense that $A\_\tau + 1 = A\_\tau$. This means that the outer measure if positive is infinite and one has to consider instead $A\_\tau \cap [0,1]$ to get something meaningful.


2) Using propositions 8.5, 8.6 and 10.4 from Falconer's book Fractal Geometry one deduces that the $2/\tau$-outer-measure of the set $\bigcap\_{\tau' < \tau} A\_{\tau'}$, which is in principle bigger than $A\_\tau$, is strictly positive.



Jarník-Besicovitch and outer measure
Given any interval $I$, the $2/\tau$-Hausdorff outer measure of the set $A\_\tau \cap I$ is always infinite for $\tau \neq 2$ and equal to $|I|$ for $\tau = 2$, independently of the constant $C$. This is a consequence of Jarník's theorem, which can be consulted here: <https://arxiv.org/abs/1601.01948>. Incidentally these notes were uploaded one month after my original question.


