Square-Root Cancellation, Averages over Hyperplanes, and the Structure of Finite Rings

We formulate a form of square-root cancellation for the operator which sums a mean-zero function over a hyperplane in $R^d$ for $R$ a possibly noncommutative finite ring. Using an argument of Hart, Iosevich, Koh, and Rudnev, we show that this square-root cancellation occurs when $R$ is a finite field. We then show that this square-root cancellation does not occur over finite rings which are not finite fields. This extends an earlier result of the author to an operator which is not translation-invariant.