Exact approximation order and well-distributed sets

Abstract

We prove that for a suitable Ahlfors regular metric measure space X and a function ψ:(0,∞)→(0,∞) from a suitable class of approximation functions, the Hausdorff dimensions of the set Wψ(Q) of all points ψ-well-approximable by a well-distributed subset Q⊂X, and the set Eψ(Q) of points that are exactly ψ-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.