In reading Chan's paper, Closest Point Problems Simplified on a RAM, the following came up as a sub-problem:
Given a set $P$ of points in the plane, and a query point $q$, find the first $k$ points (ordered by $x$-coordinate) which dominate $q$.
Chan asserts that this query can be answered in $O(\log n + k)$ time using a priority search tree, but doesn't give details, and the details are not clear to me. The traditional query for a priority search tree is a 3-sided range query, but this query is different because we are given a 2-sided range (an upper-right quadrant) and a maximum number of points to report. We should report the $k$ "leftmost" points in the 2-sided range.