Simultaneous and multiplicative Diophantine approximation on missing-digit fractals

TL;DR

This paper explores how well points on missing-digit fractals can be approximated by rational numbers, extending classical theorems to these complex fractal sets.

Contribution

It establishes analogues of Khinchin's and Gallagher's theorems for Diophantine approximation on missing-digit fractals, including inhomogeneous cases.

Findings

Proved analogues of Khinchin's theorem for missing-digit fractals

Extended Gallagher's theorem to these fractals

Developed inhomogeneous approximation results

Abstract

We investigate the metric theory of Diophantine approximation on missing-digit fractals. In particular, we establish analogues of Khinchin's theorem and Gallagher's theorem, as well as inhomogeneous generalisations.