Let $P$ denote the set of points in a slab $u$ and let $m=|P|$. We construct a $t$-shallow cutting with $t=\log^6 n$ and apply Theorem \[theor:shallowpart\] with $d=\log n$; the subset $P'\subset P$ and the set of rectangles ${{\cal R}}$ are as defined in Theorem \[theor:shallowpart\]. We keep $P'$ in a data structure from [@Chan13] that uses $O(\log n)$ bits per point and answers queries in $O(\log \log n + k)$ time. Let $P_j=list(C_j)\setminus P'$ denote the set of points in the conflict list of the cell $C_j$ that are not in $P'$. We construct a dominance data structure for points in $P_j$. Points in $P_j$ are reduced to the rank space, so that the cell data structure uses $O(\log t)=O(\log \log n)$ bits per point. A $(\log n)$-capped dominance query is answered as follows: we find the cell $C$ that contains $q$, reduce $q$ to the rank space of $C$, and report all points in $P_j$ that are dominated by $q$. We also query the data structure for $P'$ and report all points in $P'$ that are dominated by $q$. If $q$ is not contained in any cell of the $t$-shallow cutting, then $q$ dominates at least $t=\log^5 n$ 