On the quadratic complexity of subsets of $\mathbb{F}_p^n$ of bounded $\mathrm{VC_{2}}$-dimension
C. Terry, J. Wolf
TL;DR
The paper strengthens arithmetic regularity results for subsets of $\mathbb{F}_p^n$ with bounded $\mathrm{VC}_2$-dimension by improving the bound on the quadratic complexity $q$ in a quadratic factor to $q \le \log_p(\mu^{-k-o_{k}(1)})$, thereby partitioning the group into $\mu^{-k-o(1)}$ atoms at the purely quadratic level. The authors achieve this via a localized, configuration-space augmentation of the Alon–Fox–Zhao framework, combining it with prior VC$_2$-dimension regularity results to control the density on atoms and enable a pullback from a reduced configuration space to the ambient group. Central to the approach are: (i) a local stabilizer/packing argument relative to a fixed subspace $H$, (ii) a quantitative approximate local AFZ theorem, and (iii) a mechanism to translate a high-density, VC$_2$-bounded reduced structure into an IP$_2$ conclusion and a global quadratic factor with efficiently bounded complexity. A lower-bound construction shows that the improved bound on $q$ cannot be generically tightened to a constant, establishing near-optimality of the logarithmic bound in the stated regime. The results contribute toward tighter higher-order regularity theorems for subsets of finite abelian groups and have parallels in hypergraph regularity theory for bounded VC-dimension objects.
Abstract
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.