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

Abstract

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 $E=A\times \ldots \times A$ of $\mathbb{Z}_q^d$, where $A\subset \mathbb{Z}_q$.