On some fractal sets only containing irrationals

TL;DR

This paper demonstrates that certain fractal sets derived from dynamical systems exclusively contain irrational numbers, providing explicit constructions of such sets with overlapping self-similarity.

Contribution

It introduces new methods to construct and analyze fractal sets that contain only irrationals, expanding understanding of their structure and properties.

Findings

Many fractal sets from dynamical systems contain only irrationals

Explicit examples of overlapping self-similar sets with only irrationals

Enhanced understanding of the structure of such fractal sets

Abstract

In this paper, we prove that many fractal sets generated by the associated dynamical systems only contain irrationals. As an application, we explicitly construct some overlapping self-similar sets which only consist of irrationals.