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

TL;DR

This paper investigates the minimal number of kth powers needed to represent all elements in the ring of integers modulo n, providing complete solutions for exponents up to 10 and exploring related cases with applications to distance problems.

Contribution

It offers a full characterization for exponents up to 10 and discusses intermediary cases, using elementary methods and Hensel's theorem, with applications to Erdős-Falconer distance problems.

Findings

Complete solutions for k ≤ 10

Characterization of n where all elements are sums of three squares

Application to Erdős-Falconer distance problem

Abstract

We study a variant of Waring's problem for $\mathbb{Z}_n$, the ring of integers modulo $n$: For a fixed integer $k \geq 2$, what is the minimum number $m$ of $k$th powers necessary such that $x \equiv x_1^k + \dots + x_m^k \pmod{n}$ has a solution for every $x \in \mathbb{Z}_n$? Using only elementary methods, we answer fully this question for exponents $k \leq 10$, 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 $\mathbb{Z}_n^d$.