  We give an embedding of the Poincar\'e halfspace $H^D$ into a discrete metric
space based on a binary tiling of $H^D$, with additive distortion $O(\log D)$.
It yields the following results. We show that any subset $P$ of $n$ points in
$H^D$ can be embedded into a graph-metric with $2^{O(D)}n$ vertices and edges,
and with additive distortion $O(\log D)$. We also show how to construct, for
any $k$, an $O(k\log D)$-purely additive spanner of $P$ with $2^{O(D)}n$
Steiner vertices and $2^{O(D)}n \cdot \lambda_k(n)$ edges, where $\lambda_k(n)$
is the $k$th-row inverse Ackermann function. Finally, we show how to construct
an approximate Voronoi diagram for $P$ of size $2^{O(D)}n$. It allows us to
answer approximate near-neighbor queries in $2^{O(D)}+O(\log n)$ time, with
additive error $O(\log D)$. These constructions can be done in $2^{O(D)}n \log
n$ time.
