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

TL;DR

This paper investigates Erdős-type geometric problems over cyclic rings, focusing on the distribution of triangles and areas in modular integer grids, and explores dot product properties in product subsets.

Contribution

It introduces new results on geometric configurations and dot products in modular integer settings, extending classical Erdős problems to algebraic structures over cyclic rings.

Findings

Distribution results for triangles and their areas in inite cyclic rings

New bounds on dot product configurations in product subsets of inite cyclic rings

Extension of Erd
53s-type problems to algebraic structures over inite 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 $A\subset \mathbb{Z}_q$.