# On intrinsic and extrinsic rational approximation to Cantor sets

## TL;DR

This paper advances the understanding of rational approximation to fractal sets, especially Cantor sets, by establishing new bounds, extending results to multi-dimensional cases, and analyzing the structure of rational points within these fractals.

## Contribution

It introduces new approximation bounds for rational numbers near fractal sets, generalizes previous results to higher dimensions, and studies the pattern and structure of rational points in Cantor and similar fractals.

## Key findings

- Provided a lower bound for the distance of rationals outside the Cantor set to the set.
- Extended approximation results to multi-dimensional fractal sets with mild assumptions.
- Analyzed the structure of rational points in missing digit Cantor sets, including their denominator properties.

## Abstract

We establish various new results on a problem proposed by K. Mahler in 1984 concerning rational approximation to fractal sets by rational numbers inside and outside the set in question, respectively. Some of them provide a natural continuation and improvement of recent results of Broderick, Fishman and Reich and Fishman and Simmons. A key feature is that many of our new results apply to more general, multi-dimensional fractal sets and require only mild assumptions on the iterated function system. Moreover we provide a non-trivial lower bound for the distance of a rational number $p/q$ outside the Cantor middle third set $C$ to the set $C$, in terms of the denominator $q$. We further discuss patterns of rational numbers in fractal sets. We want to highlight two of them: Firstly, an upper bound for the number of rational (algebraic) numbers in a fractal set up to a given height (and degree) for a wide class of fractal sets. Secondly we find properties of the denominator structure of rational points in ''missing digit'' Cantor sets, generalizing claims of Nagy and Bloshchitsyn.