Diophantine sets in general are Cantor sets

Abstract

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

References in corpus (1)

• Isolated points of Diophantine sets