Some combinatorial number theory problems over finite valuation rings

Let $\mathcal{R}$ be a finite valuation ring of order $q^r$. In this paper we generalize and improve several well-known results, which were studied over finite fields ${\mathbb{F}}_q$ and finite cyclic rings $\mathbb{Z}\mathrm{/}{p}^r\mathbb{Z}$, in the setting of finite valuation rings.