Caroline Terry : On the structure of sets of bounded VC_2-dimension in elementary abelian p-groups

Posted by Vishesh Jain , part of the Mathematical Computer Science Seminar.

At

Nov. 3, 2025, 3 p.m.

In

1227 SEO

Abstract

We begin by presenting work of the author and Julia Wolf from 2021 showing that any subset of an elementary abelian $p$-group of bounded VC_2-dimension is well approximated by a union of atoms of a quadratic factor of bounded complexity. This result relies on a general quadratic arithmetic regularity lemma of Green and Tao, and consequently, yields bounds on the linear and quadratic complexities of the factor which are of tower-type in $\varepsilon^{-1}$, where $\varepsilon$ is the approximation parameter. We then present more recent work, also joint with Julia Wolf, which shows the bound on the quadratic complexity of the factor appearing in the structure theorem for sets of bounded VC_2-dimension can improved drastically, specifically to a logarithm in a power of $\varepsilon^{-1}$.