Mahler's question for intrinsic Diophantine approximation on triadic Cantor set: the divergence theory

TL;DR

This paper develops a complete metric theory for intrinsic Diophantine approximation on the triadic Cantor set, addressing Mahler's question by introducing a new height function based on 3-adic expansion, and establishing divergence results.

Contribution

It introduces a novel height function for rationals in the Cantor set and provides the first divergence theory for Mahler's intrinsic Diophantine approximation.

Findings

Established a complete metric divergence theory for the problem.

Introduced a height function based on 3-adic expansion.

Resolved Mahler's question for the triadic Cantor set.

Abstract

In this paper, we consider the intrinsic Diophantine approximation on the triadic Cantor set $\mathcal{K}$, i.e. approximating the points in $\mathcal{K}$ by rational numbers inside $\mathcal{K}$, a question posed by K. Mahler. By using another height function of a rational number in $\mathcal{K}$, i.e. the denominator obtained from its periodic 3-adic expansion, a complete metric theory for this variant intrinsic Diophantine approximation is presented which yields the divergence theory of Mahler's original question.