Cayley Digraphs Associated to Arithmetic Groups

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-Sarkozy 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),d2 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.