Sets of exact approximation order by rational numbers (Q1411999)

Given a decreasing function \(\psi : \mathbb R_{> 0} \to \mathbb R_{> 0}\), one defines the set of \(\psi\)-approximable real numbers \[ \mathcal K (\psi) := \left\{\xi \in \mathbb R\, ; \, \left| \xi - {p\over q}\right| < \psi (q) \text{ for infinitely many rational numbers \(p/q\)}\right\} \] and the set \[ \text{ Exact}\,(\psi) := \mathcal K (\psi) \setminus \bigcup_{m \geq 2} \mathcal K\bigl((1-m^{-1})\psi\bigr) \] of real numbers approximable at the order \(\psi\) and not better. The problem considered in this paper is to compare the ``size'' --- in the sense of Hausdorff dimension --- of these two sets. The author shows that these sets have the same Hausdorff dimension, this implies a solution to a question of Besicovitch, Dickson and Velani. The paper also contains a refinement of a theorem of Jarník (1931) concerning the Hausdorff measure of \(\mathcal K (\psi)\). The proofs are quite involved and are mainly based on the construction of tricky suitable Cantor sets.