  Let $R$ be a finite ring and define the hyperbola $H=\{(x,y) \in R \times R:
xy=1 \}$. Suppose that for a sequence of finite odd order rings of size tending
to infinity, the following "square root law" bound holds with a constant $C>0$
for all non-trivial characters $\chi$ on $R^2$: \[ \left| \sum_{(x,y)\in
H}\chi(x,y)\right|\leq C\sqrt{|H|}. \] Then, with a finite number of
exceptions, those rings are fields.
  For rings of even order we show that there are other infinite families given
by Boolean rings and Boolean twists which satisfy this square-root law
behavior. We classify the extremal rings, those for which the left hand side of
the expression above satisfies the worst possible estimate. We also describe
applications of our results to problems in graph theory and geometric
combinatorics.
  These results provide a quantitative connection between the square root law
in number theory, Salem sets, Kloosterman sums, geometric combinatorics, and
the arithmetic structure of the underlying rings.
