Metric Diophantine approximation on fractals

TL;DR

This paper investigates how well irrational numbers within missing digit sets can be approximated by rationals with polynomial denominators, extending Khintchine's theorem to fractal-like sets.

Contribution

It establishes a Khintchine-like convergence theorem for missing digit sets with large bases, advancing understanding of Diophantine approximation on fractals.

Findings

Proves a convergence theorem for missing digit sets.

Extends Khintchine's theorem to fractal sets.

Provides bounds for approximation quality in fractal contexts.

Abstract

Inspired by a problem proposed by Mahler, we will address the following related question, 'How well can irrationals in a missing digit set be approximated by rationals with polynomial denominators?' and prove some related results. To achieve this, we will be closely looking at Khintchine's theorem, particularly the convergence case and aim to prove a Khintchine-like convergence theorem for missing digit sets with large bases and rationals with polynomial denominators.