(This talk is based on joint work with M. Bennett, D. Hart, A. Iosevich and M. Rudnev.)
In this talk, we will discuss how the methodology of group actions and stabilizer subgroups can streamline Fourier analytic proofs in extremal combinatorics over finite fields. In particular, we obtain lower bounds on the size of a subset of a vector space over a finite field which guarantee that it represents a positive proportion of possible congruence (or similarity) types of k-simplicies where k is less than or equal to the ambient dimension d of the vector space. This includes bounds for when a subset generates "most" distances, triangles, tetrahedra and so on.
This particular method generalizes and has obtained better bounds in the critical case k=d (and also k=d-1) then previous known results. It also lends itself to yielding better results for subsets of spheres than can be obtained by applying results for vector spaces, i.e. it yields an effective tool to exploit the additional geometry present for subsets constrained to a sphere. This group theoretic approach leads to formulae in the style of the classical Mattila integral from geometric measure theory and serve as an effective analog.
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