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

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 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 $\mathrm{liminf}$ set and combine this with ideas inspired by the proof of the Mass Transference Principle to find a large $\mathrm{limsup}$ subset of the $\mathrm{liminf}$ 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.

25 pages