The structure of subsets of $\mathbb{F}_p^n$ of bounded $\mathrm{VC}_2$-dimension
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| Format: | Preprint |
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2025
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| _version_ | 1866908597869346816 |
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| author | Terry, C. Wolf, J. |
| author_facet | Terry, C. Wolf, J. |
| contents | We show that a subset of $\mathbb{F}_{p}^{n}$ of $\mathrm{VC_{2}}$-dimension at most $k$ is well approximated by a union of atoms of a quadratic factor of complexity $(\ell,q)$ (denoting the complexities of the linear and quadratic part, respectively), where $\ell$ and $q$ are bounded by a constant depending only on $k$ and the desired level of approximation. This generalises a result of Alon, Fox and Zhao on the structure of sets of bounded $\mathrm{VC}$-dimension, and is analogous to contemporaneous work of the authors arXiv:2111.01737 in the setting of 3-uniform hypergraphs.
The main result originally appeared--albeit with a different proof--in a 2021 preprint arXiv:2111.01739, which has since been split into two: the present work, which focuses on higher arity NIP and develops a theory of local uniformity semi-norms of possibly independent interest, and its companion arXiv:2111.01739, which strengthens these results under a generalized notion of stability. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_12867 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The structure of subsets of $\mathbb{F}_p^n$ of bounded $\mathrm{VC}_2$-dimension Terry, C. Wolf, J. Combinatorics Logic We show that a subset of $\mathbb{F}_{p}^{n}$ of $\mathrm{VC_{2}}$-dimension at most $k$ is well approximated by a union of atoms of a quadratic factor of complexity $(\ell,q)$ (denoting the complexities of the linear and quadratic part, respectively), where $\ell$ and $q$ are bounded by a constant depending only on $k$ and the desired level of approximation. This generalises a result of Alon, Fox and Zhao on the structure of sets of bounded $\mathrm{VC}$-dimension, and is analogous to contemporaneous work of the authors arXiv:2111.01737 in the setting of 3-uniform hypergraphs. The main result originally appeared--albeit with a different proof--in a 2021 preprint arXiv:2111.01739, which has since been split into two: the present work, which focuses on higher arity NIP and develops a theory of local uniformity semi-norms of possibly independent interest, and its companion arXiv:2111.01739, which strengthens these results under a generalized notion of stability. |
| title | The structure of subsets of $\mathbb{F}_p^n$ of bounded $\mathrm{VC}_2$-dimension |
| topic | Combinatorics Logic |
| url | https://arxiv.org/abs/2510.12867 |