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We explore a paradigm which ties together seemingly disparate areas in number theory, additive combinatorics, and geometric combinatorics including the classical Waring problem, the Furstenberg-Sárközy theorem on squares in sets of integers with positive density, and the study of triangles (also called 2-simplices) in finite fields. Among other results we show that if Fq is the finite field of odd order q, then every matrix in Matd(Fq),d≥2 is the sum of a certain (finite) number of orthogonal matrices, this number depending only on d, the size of the matrix, and on whether q is congruent to 1 or 3 (mod 4), but independent of q otherwise. [ABSTRACT FROM AUTHOR]

In this paper we study the extremal type problem arising from the question: What is the maximum number ET( S) of edges that a geometric graph G on a planar point set S can have such that it does not contain empty triangles? We prove: $${{n \choose 2} - O(n \log n) \leq ET(n) \leq {n \choose 2} - \left(n - 2 + \left\lfloor \frac{n}{8} \right\rfloor \right)}$$ . [ABSTRACT FROM AUTHOR]