  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-S\'{a}rk\"{o}zy 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
$\mathbb{F}_q$ is the finite field of odd order $q$, then every matrix in
$Mat_d(\mathbb{F}_q), d \geq 2$ 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.
