Higher-order generalizations of stability and arithmetic regularity
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866912652440109056 |
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| author | Terry, C. Wolf, J. |
| author_facet | Terry, C. Wolf, J. |
| contents | We define a natural notion of higher order stability and show that subsets of $\mathbb{F}_p^n$ that are tame in this sense can be approximately described by a union of low-complexity quadratic varieties, up to linear error. This generalizes the arithmetic regularity lemma for stable subsets of $\mathbb{F}_p^n$, proved in earlier work of the authors, to the realm of higher-order Fourier analysis.
This result is strictly stronger than the structure theorem for sets of bounded $\mathrm{VC}_2$-dimension, first proved by the authors in earlier versions of this paper and now available as a separate manuscript arXiv:2510.12867. Taken together, these results provide group theoretic analogues of results obtained for 3-uniform hypergraphs in arXiv:2111.01737. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_01739 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Higher-order generalizations of stability and arithmetic regularity Terry, C. Wolf, J. Combinatorics Logic We define a natural notion of higher order stability and show that subsets of $\mathbb{F}_p^n$ that are tame in this sense can be approximately described by a union of low-complexity quadratic varieties, up to linear error. This generalizes the arithmetic regularity lemma for stable subsets of $\mathbb{F}_p^n$, proved in earlier work of the authors, to the realm of higher-order Fourier analysis. This result is strictly stronger than the structure theorem for sets of bounded $\mathrm{VC}_2$-dimension, first proved by the authors in earlier versions of this paper and now available as a separate manuscript arXiv:2510.12867. Taken together, these results provide group theoretic analogues of results obtained for 3-uniform hypergraphs in arXiv:2111.01737. |
| title | Higher-order generalizations of stability and arithmetic regularity |
| topic | Combinatorics Logic |
| url | https://arxiv.org/abs/2111.01739 |