  Let $\gamma\in(0;\frac{1}{2}),\tau\geq 1$ and define the "$\gamma,\tau$
Diophantine set" as: $$D_{\gamma,\tau}:=\{\alpha\in (0;1):
||q\alpha||\geq\frac{\gamma}{q^{\tau}}\quad\forall
q\in\Bbb{N}\},\qquad||x||:=\inf_{p\in\Bbb{Z}}|x-p|. $$ In this paper we study
the topology of these sets and we show that, for large $\tau$ and for almost
all $\gamma>0$, $D_{\gamma,\tau}$ is a Cantor set.
