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

Abstract

Let $ψ:\mathbb{N} \to [0,\infty)$, $ψ(q)=q^{-(1+τ)}$ and let $ψ$-badly approximable points be those vectors in $\mathbb{R}^{d}$ that are $ψ$-well approximable, but not $cψ$-well approximable for arbitrarily small constants $c>0$. We establish that the $ψ$-badly approximable points have the Hausdorff dimension of the $ψ$-well approximable points, the dimension taking the value $(d+1)/(τ+1)$ familiar from theorems of Besicovitch and Jarník. 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.