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–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.