The Dimension of the Set of $\psi $-Badly Approximable Points in All Ambient Dimensions: On a Question of Beresnevich and Velani

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 Jarník. The method of proof is an entirely new take on the Mass Transference Principle (MTP) 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 MTP 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.