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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 arbitrarilysmall constants $c>0$. We establish that the $\psi$-badly approximable pointshave the Hausdorff dimension of the $\psi$-well approximable points, thedimension taking the value $(d+1)/(\tau+1)$ familiar from theorems ofBesicovitch and Jarn\'ik. The method of proof is an entirely new take on theMass Transference Principle by Beresnevich and Velani (Annals, 2006); namely,we use the colloquially named `delayed pruning' to construct a sufficientlylarge $\liminf$ set and combine this with ideas inspired by the proof of theMass Transference Principle to find a large $\limsup$ subset of the $\liminf$set. Our results are a generalisation of some $1$-dimensional results due toBugeaud and Moreira (Acta Arith, 2011), but our method of proof is nothingalike.

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