Geometric combinatorics in discrete settings
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This thesis is a compilation of work in which the author studies geometric configurations in finite fields and the integers modulo q. The results of this dissertation are threefold. First, we prove a finite field analog of the Furstenberg-Katznelson-Weiss theorem on triangles in R2. Second, we study volume sets in Fd/q and discuss some applications to sum-product problems. Finally, we study geometric combinatorics in Z/qZ. We generalize a result of Hart and Iosevich  which has applications to sum-product problems. Finally, we show that the Zd/q analogue of a sphere with unital radius is qd−1-dimensional.