The structure of subsets of $\mathbb{F}_p^n$ of bounded $\mathrm{VC}_2$-dimension

Abstract

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.