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arXiv · 2504.00363

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

Nathaniel Kingsbury-Neuschotz
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Abstract

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

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Nathaniel Kingsbury-Neuschotz. 2026-06-29. Square-Root Cancellation, Averages over Hyperplanes, and the Structure of Finite Rings. https://arxiv.org/abs/2504.00363

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