On the quadratic complexity of subsets of $\mathbb{F}_p^n$ of bounded $\mathrm{VC_{2}}$-dimension
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2025
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| author | Terry, C. Wolf, J. |
| author_facet | Terry, C. Wolf, J. |
| contents | In prior work, we showed that subsets of $\mathbb{F}_{p}^{n}$ of $\mathrm{VC_{2}}$-dimension at most $k$ are well approximated by a union of atoms of a quadratic factor of complexity $(\ell,q)$, where the complexity $\ell$ of the linear part and the complexity $q$ of the quadratic part are both bounded in terms of $k$, $p$, and the desired level of approximation $μ$. A key tool in the proof of this result was an arithmetic regularity lemma for the Gowers $U^3$-norm by Green and Tao, which resulted in tower-type bounds (in terms of $μ^{-1}$) on both $\ell$ and $q$.
In the present paper we show that for sets of bounded $\mathrm{VC}_2$-dimension, the bound on $q$ can be substantially improved. Specifically, we will prove that any set $A\subseteq G=\mathbb{F}_p^n$ of $\mathrm{VC}_2$-dimension at most $k$ is approximately equal (up to error $μ|G|$) to a union of atoms of a quadratic factor whose quadratic complexity is at most $\log_p(μ^{-k-o(1)})$, implying that the purely quadratic component of the factor partitions the group into $μ^{-k-o(1)}$ many parts.
We achieve this by using our earlier result to obtain an initial quadratic factor $\mathcal{B}$, and then applying a generalization of an argument of Alon, Fox and Zhao for subsets of $\mathbb{F}_{p}^{n}$ of bounded $\mathrm{VC}$-dimension to the label space (also known as "configuration space") of $\mathcal{B}$. A related strategy was employed in earlier work of the authors on $\mathrm{NFOP}_2$ subsets of $\mathbb{F}_p^n$, and in work of the first author in the context of 3-uniform hypergraphs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_12767 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the quadratic complexity of subsets of $\mathbb{F}_p^n$ of bounded $\mathrm{VC_{2}}$-dimension Terry, C. Wolf, J. Combinatorics Logic In prior work, we showed that subsets of $\mathbb{F}_{p}^{n}$ of $\mathrm{VC_{2}}$-dimension at most $k$ are well approximated by a union of atoms of a quadratic factor of complexity $(\ell,q)$, where the complexity $\ell$ of the linear part and the complexity $q$ of the quadratic part are both bounded in terms of $k$, $p$, and the desired level of approximation $μ$. A key tool in the proof of this result was an arithmetic regularity lemma for the Gowers $U^3$-norm by Green and Tao, which resulted in tower-type bounds (in terms of $μ^{-1}$) on both $\ell$ and $q$. In the present paper we show that for sets of bounded $\mathrm{VC}_2$-dimension, the bound on $q$ can be substantially improved. Specifically, we will prove that any set $A\subseteq G=\mathbb{F}_p^n$ of $\mathrm{VC}_2$-dimension at most $k$ is approximately equal (up to error $μ|G|$) to a union of atoms of a quadratic factor whose quadratic complexity is at most $\log_p(μ^{-k-o(1)})$, implying that the purely quadratic component of the factor partitions the group into $μ^{-k-o(1)}$ many parts. We achieve this by using our earlier result to obtain an initial quadratic factor $\mathcal{B}$, and then applying a generalization of an argument of Alon, Fox and Zhao for subsets of $\mathbb{F}_{p}^{n}$ of bounded $\mathrm{VC}$-dimension to the label space (also known as "configuration space") of $\mathcal{B}$. A related strategy was employed in earlier work of the authors on $\mathrm{NFOP}_2$ subsets of $\mathbb{F}_p^n$, and in work of the first author in the context of 3-uniform hypergraphs. |
| title | On the quadratic complexity of subsets of $\mathbb{F}_p^n$ of bounded $\mathrm{VC_{2}}$-dimension |
| topic | Combinatorics Logic |
| url | https://arxiv.org/abs/2510.12767 |