Diophantine sets in general are Cantor sets

Let $\gamma \in (0;\frac{1}{2}),\tau \ge 1$ and define the “$\gamma ,\tau$ Diophantine set” as: $$D_{\gamma ,\tau }:=\{\alpha \in (0;1):\mathrm{\mid }\mathrm{\mid }q\alpha \mathrm{\mid }\mathrm{\mid }\ge \frac{\gamma }{q^{\tau }}\mathrm{\forall }q\in \mathbb{N}\},\mathrm{\mid }\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }:=\underset{p\in \mathbb{Z}}{\inf }\mathrm{\mid }x-p\mathrm{\mid }\mathrm{.}$$ 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.