Isolated points of Diophantine sets

TL;DR

This paper investigates the topological structure of certain Diophantine sets defined by inequalities involving approximation quality, revealing that these sets typically contain isolated points.

Contribution

It provides a detailed analysis of the topology of $ ext{Diophantine sets}$, demonstrating that they generally possess isolated points, which was previously not well understood.

Findings

Diophantine sets often have isolated points

Topological properties depend on parameters $eta$ and $ au$

The sets are not necessarily perfect or continuous

Abstract

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