The square root law and structure of finite rings

TL;DR

This paper investigates the structure of finite rings satisfying a 'square root law' bound on character sums over hyperbolas, showing most are fields or special constructions, with applications to graph theory and combinatorics.

Contribution

It classifies finite rings satisfying the square root law, identifies extremal cases, and explores connections to number theory, combinatorics, and geometric problems.

Findings

Most large odd-order rings satisfying the law are fields.

Boolean rings and twists satisfy the law for even-order rings.

Character sum bounds relate to geometric and combinatorial structures.

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.