Exact approximation order and well-distributed sets (Q2685689)

One can begin with authors' abstract: ``We prove that for suitable Ahlfors regular metric measure space \(X\) and a function \(\psi:(0,\infty)\to(0,\infty)\) from a suitable class of approximation functions, the Hausdorff dimensions of the set \(W_\psi(Q)\) of all points \(\psi\)-well-approximable by a well-distributed subset \(Q\subset X\), and the set \(E_\psi(Q)\) of points that are exactly \(\psi\)-approximable by \(Q\), coincide. This answers in a general setting, a question of Beresnevich-Dickinson-Velani in the case of approximation of reals by rationals, and answered by Bugeaud in that case using the continued-fraction expansion of reals. Our main result applies in particular to approximation by orbits of fixed points of a wide class of discrete groups of isometries acting on the boundary of hyperbolic metric spaces.'' Special attention is given to a brief survey on certain results which are related to Diophantine approximations and the Hausdorff dimension of ``exactly approximable'' sets. Also, the authors explain the motivation of these investigations and give some remarks and examples.