  This paper studies empty squares in arbitrary orientation among a set $P$ of
$n$ points in the plane. We prove that the number of empty squares with four
contact pairs is between $\Omega(n)$ and $O(n^2)$, and that these bounds are
tight, provided $P$ is in a certain general position. A contact pair of a
square is a pair of a point $p\in P$ and a side $\ell$ of the square with $p\in
\ell$. The upper bound $O(n^2)$ also applies to the number of empty squares
with four contact points, while we construct a point set among which there is
no square of four contact points. These combinatorial results are based on new
observations on the $L_\infty$ Voronoi diagram with the axes rotated and its
close connection to empty squares in arbitrary orientation. We then present an
algorithm that maintains a combinatorial structure of the $L_\infty$ Voronoi
diagram of $P$, while the axes of the plane continuously rotates by $90$
degrees, and simultaneously reports all empty squares with four contact pairs
among $P$ in an output-sensitive way within $O(s\log n)$ time and $O(n)$ space,
where $s$ denotes the number of reported squares. Several new algorithmic
results are also obtained: a largest empty square among $P$ and a square
annulus of minimum width or minimum area that encloses $P$ over all
orientations can be computed in worst-case $O(n^2 \log n)$ time.
