On the linear complexity of subsets of $\mathbb{F}_p^n$ bounded $\textrm{VC}_2$-dimension

Previous work of the second author and Wolf showed that given a set $A\subseteq \mathbb{F}_p^n$ of bounded $\textrm{VC}_2$-dimension, there is a high rank quadratic factor $\mathcal{B}$ of bounded complexity such that $A$ is approximately equal to a union of atoms of $\mathcal{B}$. That proof yielded bounds of tower type on the linear and quadratic complexities. It was later shown by the same authors that the quadratic complexity can be improved to logarithmic, however that proof provided no improvement on the linear component. In this paper we prove that the bound on the linear complexity can be improved to a triple exponential in the case of linear rank functions, and a quadruple exponential for polynomial rank functions of higher degree. Our strategy is based on the one developed by Gishboliner, Wigderson, and Shapira to prove the analogous result in the hypergraph setting. Step 1 is to prove a``cylinder" version of the quadratic arithmetic regularity lemma, which says that given a set $A\subseteq G=\mathbb{F}_p^n$, there is a partition of $G$ into atoms of (possibly distinct) quadratic factors of high rank and bounded complexity, so that most atoms in the partition are uniform with respect to the set $A$, in the sense of a certain local $U^3$ norm. Step 2 is to show that if $A$ has bounded $\textrm{VC}_2$-dimension, then it has density near $0$ or $1$ on all atoms which are uniform in the sense of Step 1. Step 1 relies on a recent local version of the $U^3$ inverse theorem due to Prendiville, and is necessarily phrased in terms of a local $U^3$ norm implicit in that paper. On the other hand, Step 2 relies on a counting lemma for a different local $U^3$ due to Terry and Wolf, which we prove here is approximately the same as the local $U^3$ norm used in Step 1.