Square-Root Cancellation, Averages over Hyperplanes, and the Structure of Finite Rings
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909709732151296 |
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| author | Kingsbury-Neuschotz, Nathaniel |
| author_facet | Kingsbury-Neuschotz, Nathaniel |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_00363 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Square-Root Cancellation, Averages over Hyperplanes, and the Structure of Finite Rings Kingsbury-Neuschotz, Nathaniel Number Theory Classical Analysis and ODEs Combinatorics 11L05 (Primary) 52C10, 44A12 (Secondary) 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. |
| title | Square-Root Cancellation, Averages over Hyperplanes, and the Structure of Finite Rings |
| topic | Number Theory Classical Analysis and ODEs Combinatorics 11L05 (Primary) 52C10, 44A12 (Secondary) |
| url | https://arxiv.org/abs/2504.00363 |