Erd\H{o}s Type Problems in Modules over Cyclic Rings

In the present paper, we study various Erd\H{o}s type geometric problems in the setting of the integers modulo $q$, where $q=p^l$ is an odd prime power. More precisely, we prove certain results about the distribution of triangles and triangle areas among the points of $E\subset {\mathbb{Z}}_q^2$. We also prove a dot product result for $d$-fold product subsets $E=A\times \dots \times A$ of ${\mathbb{Z}}_q^d$, where $A\subset {\mathbb{Z}}_q$.