Exact approximation order and well-distributed sets

We prove that for any proper metric space $X$ and a function $ψ:(0,\infty)\to(0,\infty)$ from a suitable class of approximation functions, the Hausdorff dimensions of the set $W_ψ(Q)$ of all points $ψ$-well-approximable by a well-distributed subset $Q\subset X$, and the set $E_ψ(Q)$ 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.

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