stores the points of the input set $P$ sorted by their fourth coordinate. In every internal node $u$ of the range tree, we keep a set of points $S(u)$; $S(u)$ contains all points stored in the leaf descendants of $u$. In every node $u$ we keep a data structure supporting three-dimensional dominance queries. Let $\gamma$ denote the node degree of the range tree. For any interval $[a,b]$, we can identify $O(\gamma \cdot \log_{\gamma} n)$ nodes $u_i$ of the range tree, such that $p.z'\in [a,b]$ if and only if $p\in S(u_i)$. Furthermore nodes $u_i$ can be divided into $O(\log_{\gamma}n)$ groups, such that nodes in the same group are siblings.

A three-dimensional dominance query can be answered by locating a point in a planar orthogonal subdivision [@Nekrich07; @Afshani08; @Chan13]. Orthogonal point location queries can be answered in $O(\log \log n)$ time [@Chan13]. If the range tree $T$ is a binary tree, we can answer four-dimensional dominance queries in $O(\log n \log \log n + k)$ time. By increasing the node degree $\gamma$ to $O(\log^{{\varepsilon}}n)$, we can reduce the query time to $O(\log n + k)$. Unfortunately it appears that further improvement in query time is not possible with this approach: $O(\log \log 