Waring's Problem in Finite Rings

Cayley Digraphs Associated to Arithmetic Groups

“…The determination of the diameter of T G is thus a variant of Waring's problem over F q . The solution of the Waring problem for general (not necessarily commutative) finite rings was obtained by the second listed author in [4] in part by studying this graph and its spectrum.…”

“…Recall that the classical Furstenberg-Sárközy Theorem ( [7,13]) states that every set of integers with positive (natural) density contains two elements whose difference is a square. The spectral theorem applied to these power subgroups G = (F * q ) k graphs also yields ( [4]) a finite field generalization of the Furstenberg-Sárközy Theorem for all powers k ≥ 2: if E ⊆ F q satisfies |E| ≫ k √ q then there exists e, e ′ ∈ E, a ∈ F q with e − e ′ = a k . The question of determining the minimum number, say m, of units such that every element is a sum of m units is well known (see [14], for example).…”