  Given a set $P$ of $n$ points in the plane, we consider the problem of
computing the number of points of $P$ in a query unit disk (i.e., all query
disks have the same radius). We show that the main techniques for simplex range
searching in the plane can be adapted to this problem. For example, by adapting
Matou\v{s}ek's results, we can build a data structure of $O(n)$ space so that
each query can be answered in $O(\sqrt{n})$ time. Our techniques lead to
improvements for several other classical problems, such as batched range
searching, counting/reporting intersecting pairs of unit circles, distance
selection, discrete 2-center, etc. For example, given a set of $n$ unit disks
and a set of $n$ points in the plane, the batched range searching problem is to
compute for each disk the number of points in it. Previous work [Katz and
Sharir, 1997] solved the problem in $O(n^{4/3}\log n)$ time while our new
algorithm runs in $O(n^{4/3})$ time.
