Exact approximation order and well-distributed sets

Abstract: We prove that for any proper metric 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.