The Square-Root Law Does Not Hold in the Presence of Zero Divisors

Nathaniel Kingsbury-Neuschotz

Let $R$ be a finite ring (with unit, not necessarily commutative) and define the paraboloid $P=\{(x_1,\dots ,x_d)\in R^d|x_d=x_1^2+\cdots +x_{d-1}^2\}.$ Suppose that for a s…