Title: The structure of sets of bounded VC_2-dimension
Abstract: VC-dimension is a classical measure of combinatorial tameness, widely used across model theory, statistics, combinatorics, and theoretical computer science since its inception in the 1970s. A higher-dimensional analogue, known as VC_2-dimension, was introduced by Shelah in 2014 but has thus far received comparatively little attention.
In joint work with Caroline Terry, we showed that subsets of bounded VC_2-dimension in a finite elementary abelian p-group can be approximated by a union of quadratic “atoms”, that is, simultaneous level sets of a bounded number of high-rank quadratic and a bounded number of linear forms—the basic structured objects of quadratic Fourier analysis. This results generalises prior work of Alon-Fox-Zhao and Sisask for subsets of bounded VC-dimension.
Recent joint work with Terry, along with work of Sheats and Terry, on the quantitative aspects of this problem sheds new light on some previously poorly understood aspects of quadratic Fourier analysis.
Rachel Greenfeld
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