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

Let ψ : N → [0, ∞), ψ(q) = q−(1+τ) and let ψ-badly approximable points be those vectors in Rd 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 (MTP) by Beresnevich and Velani (Annals, 2006); namely, we use the colloquially named “delayed pruning” to construct a sufficiently large lim inf set and combine this with ideas inspired by the proof of the MTP to find a large lim sup subset of the lim inf 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.