Point configurations over finite fields
Parshall, Hans David
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We model classical problems in Euclidean Ramsey theory by considering geometric structure that must appear within large, but otherwise arbitrary, subsets of vector spaces over finite fields. Point configurations in general position are located using Fourier analytic arguments, while higher complexity point configurations require the development of geometric Gowers-style uniformity norms. We obtain new results for low complexity configurations and outline a strategy to locate arbitrary spherical configurations.