  We introduce dynamic smooth (a.k.a. balanced) compressed quadtrees with
worst-case constant time updates in constant dimensions. We distinguish two
versions of the problem. First, we show that quadtrees as a space-division data
structure can be made smooth and dynamic subject to split and merge operations
on the quadtree cells. Second, we show that quadtrees used to store a set of
points in $\mathbb{R}^d$ can be made smooth and dynamic subject to insertions
and deletions of points. The second version uses the first but must
additionally deal with compression and alignment of quadtree components. In
both cases our updates take $2^{\mathcal{O}(d\log d )}$ time, except for the
point location part in the second version which has a lower bound of $\Theta
(\log n)$---but if a pointer (finger) to the correct quadtree cell is given,
the rest of the updates take worst-case constant time. Our result implies that
several classic and recent results (ranging from ray tracing to planar point
location) in computational geometry which use quadtrees can deal with arbitrary
point sets on a real RAM pointer machine.
