Abstract
In this paper, we use Cayley digraphs to obtain some new self-contained proofs for Waring’s problem over finite fields, proving that any element of a finite field Fq can be written as a sum of m many kth powers as long as q>k2mm−1; and we also compute the smallest positive integers m such that every element of Fq can be written as a sum of m many kth powers for all q too small to be covered by the above mentioned results when 2⩽k⩽37.
In the process of developing the proofs mentioned above, we arrive at another result (providing a finite field analogue of Furstenberg–Sárközy’s Theorem) showing that any subset E of a finite field Fq for which |E|>qkq−1 must contain at least two distinct elements whose difference is a kth power.