  We study geometric set cover problems in dynamic settings, allowing
insertions and deletions of points and objects. We present the first dynamic
data structure that can maintain an $O(1)$-approximation in sublinear update
time for set cover for axis-aligned squares in 2D. More precisely, we obtain
randomized update time $O(n^{2/3+\delta})$ for an arbitrarily small constant
$\delta>0$. Previously, a dynamic geometric set cover data structure with
sublinear update time was known only for unit squares by Agarwal, Chang, Suri,
Xiao, and Xue [SoCG 2020]. If only an approximate size of the solution is
needed, then we can also obtain sublinear amortized update time for disks in 2D
and halfspaces in 3D. As a byproduct, our techniques for dynamic set cover also
yield an optimal randomized $O(n\log n)$-time algorithm for static set cover
for 2D disks and 3D halfspaces, improving our earlier $O(n\log n(\log\log
n)^{O(1)})$ result [SoCG 2020].
