The dimension of the set of $\psi$-badly approximable points in all ambient dimensions; on a question of Beresnevich and Velani

TL;DR

This paper proves that the Hausdorff dimension of $ ext{psi}$-badly approximable points in $ ext{R}^d$ matches that of $ ext{psi}$-well approximable points, using a novel approach to the Mass Transference Principle.

Contribution

It introduces a new method called 'delayed pruning' to analyze the dimension of $ ext{psi}$-badly approximable points, extending previous 1D results to higher dimensions.

Findings

Hausdorff dimension of $ ext{psi}$-badly approximable points equals $(d+1)/( au+1)$

New proof technique for Mass Transference Principle involving 'delayed pruning'

Generalizes 1D results to all ambient dimensions.

Abstract

Let $\psi:\mathbb{N} \to [0,\infty)$, $\psi(q)=q^{-(1+\tau)}$ and let $\psi$-badly approximable points be those vectors in $\mathbb{R}^{d}$ that are $\psi$-well approximable, but not $c\psi$-well approximable for arbitrarily small constants $c>0$. We establish that the $\psi$-badly approximable points have the Hausdorff dimension of the $\psi$-well approximable points, the dimension taking the value $(d+1)/(\tau+1)$ familiar from theorems of Besicovitch and Jarn\'ik. The method of proof is an entirely new take on the Mass Transference Principle by Beresnevich and Velani (Annals, 2006); namely, we use the colloquially named `delayed pruning' to construct a sufficiently large $\liminf$ set and combine this with ideas inspired by the proof of the Mass Transference Principle to find a large $\limsup$ subset of the $\liminf$ set. Our results are a generalisation of some $1$-dimensional results due to Bugeaud and Moreira (Acta Arith, 2011), but our method of proof is nothing alike.