| 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 Fdq and discuss some applications to sum-product problems. Finally, we study geometric combinatorics in Z/qZ . We generalize a result of Hart and Iosevich [27] which has applications to sum-product problems. Finally, we show that the Zdq analogue of a sphere with unital radius is qd --1-dimensional. |
