Weighted approximation in higher-dimensional missing digit sets

TL;DR

This paper applies the mass transference principle to analyze the Hausdorff dimension of weighted well-approximable points in higher-dimensional missing digit sets, extending classical results to complex fractal structures.

Contribution

It introduces a novel approach using the mass transference principle for rectangles to study weighted approximation in higher-dimensional self-similar sets.

Findings

Determined Hausdorff dimension of weighted well-approximable points in missing digit sets.

Extended classical Diophantine approximation results to higher-dimensional fractals.

Applied recent mass transference principles to complex self-similar structures.

Abstract

In this note, we use the mass transference principle for rectangles, recently obtained by Wang and Wu (Math. Ann., 2021), to study the Hausdorff dimension of sets of "weighted $\Psi$-well-approximable" points in certain self-similar sets in $\mathbb{R}^{d}$. Specifically, we investigate weighted $\Psi$-well-approximable points in "missing digit" sets in $\mathbb{R}^{d}$. The sets we consider are natural generalisations of Cantor-type sets in $\mathbb{R}$ to higher dimensions and include, for example, four corner Cantor sets (or Cantor dust) in the plane with contraction ratio $\frac{1}{n}$ with $n \in \mathbb{N}$.