Isolated points of Diophantine sets

Abstract

Let $\gamma \in (0;\frac{1}{2}),\tau \ge 1$ and define the "$\gamma ,\tau$ Diophantine set" as: $$D_{\gamma ,\tau }:=\{\alpha \in (0;1):||q\alpha ||\ge \frac{\gamma }{q^{\tau }}\forall q\in \mathbb{N}\},||x||:=\underset{p\in \mathbb{Z}}{\inf }|x-p|.$$ We analyze the topology of these sets and we show that generally they have isolated points.

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• Diophantine sets in general are Cantor sets