An analogue of Furstenberg–Sárközy’s theorem and an alternative solution to Waring’s problem over finite fields

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 $F_q$ can be written as a sum of $m$ many $k\text{th}$ powers as long as $q>k^{\frac{2m}{m-1}}$; and we also compute the smallest positive integers $m$ such that every element of $F_q$ can be written as a sum of $m$ many $k\text{th}$ 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 $F_q$ for which $\left|E\right|>\frac{qk}{\sqrt{q-1}}$ must contain at least two distinct elements whose difference is a $k\text{th}$ power.