Diophantine sets in general are Cantor sets

TL;DR

This paper investigates the topological structure of Diophantine sets defined by approximation conditions and demonstrates that, under certain parameters, these sets are Cantor sets for almost all values of a parameter.

Contribution

It proves that for large au and almost all \,gamma, the Diophantine sets are Cantor sets, revealing their fractal nature.

Findings

Diophantine sets are Cantor sets for large au and almost all \,gamma.

The topology of these sets is studied and characterized.

Results contribute to understanding the fractal structure of Diophantine approximation sets.

Abstract

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.