Square-Root Cancellation, Averages over Hyperplanes, and the Structure of Finite Rings
Number Theory
2025-07-30 v2 Classical Analysis and ODEs
Combinatorics
Abstract
We formulate a form of square-root cancellation for the operator which sums a mean-zero function over a hyperplane in for a possibly noncommutative finite ring. Using an argument of Hart, Iosevich, Koh, and Rudnev, we show that this square-root cancellation occurs when 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.
Cite
@article{arxiv.2504.00363,
title = {Square-Root Cancellation, Averages over Hyperplanes, and the Structure of Finite Rings},
author = {Nathaniel Kingsbury-Neuschotz},
journal= {arXiv preprint arXiv:2504.00363},
year = {2025}
}
Comments
23 pages