Recently I discovered that given a finite field $\mathbb{F}_q$ with $q = p^m$ and $H \leq \mathbb{F}_q$ an affine subspace then $\sum_{a \in H} a^n = 0$ for all $n < |H|-2$. This sounds like an intereseting property and I was curious to see if there is any analogous result over ring.
To be more specific I'm intereseted in the case where $R$ is Galois Ring, defined as $\displaystyle \mathbb{Z}_{p^k}[x]/f(x)$ with $f(x)$ irreducible over $\mathbb{F}_p$. This ring have many properties in common with its residue field $\mathbb{F}_{p^k}$ one of which is the existence of sets $E \subseteq R$ (called exceptional) over which it is possibile to interpolate polynomials. My question is if there exists any known construction/result about exceptional sets that satisfies the property $\sum_{a \in E} a^n = 0$ for all $n \leq |E| - 2$