Geometric configurations in the ring of integers modulo $p^{\ell}$

TL;DR

This paper investigates geometric problems like the Erdős distance and dot product problems within the ring of integers modulo a prime power, extending classical finite field results to a modular setting.

Contribution

It introduces new variants of geometric problems in the ring of integers modulo p^l, broadening the scope of finite field geometric combinatorics.

Findings

Extended Erdős distance problem to modular rings

Analyzed dot product configurations in modular settings

Provided bounds and structural results for geometric configurations

Abstract

We study variants of the Erd\H os distance problem and dot products problem in the setting of the integers modulo $q$, where $q = p^{\ell}$ is a power of an odd prime.