A variant of Waring's Problem for the ring of integers modulo n

We study a variant of Waring’s problem for ${\mathbb{Z}}_n$, the ring of integers modulo $n$: For a fixed integer $k\ge 2$, what is the minimum number $m$ of $k$th powers necessary such that $x \equiv x_1^k + \dots + x_m^k \pmod{n}$hasasolutionforevery$x \in $\mathbb{Z}$_n$? Using only elementary methods, we answer fully this question for exponents $k\le 10$, and we further discuss some intermediary cases such as categorizing the values of $n$ such that every element in ${\mathbb{Z}}_n$ can be written as a sum of three squares. Hensel’s Theorem for $p$-adic integers plays a key role. Finally, we give an application of this problem to the Erd\H os-Falconer distance problem for rings ${\mathbb{Z}}_n^d$.