The square root law and structure of finite rings

Abstract: 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.