Integral point sets over $\mathbb{Z}_n^m$

TL;DR

This paper explores the properties of point sets in modular integer spaces, focusing on distances and coordinates within the finite ring rac{}n, advancing combinatorial understanding in modular geometry.

Contribution

It introduces the study of point sets over rac{}n^m, analyzing their properties and differences from classical Euclidean and integer grid cases.

Findings

Characterization of point sets in rac{}n^m

Comparison with Euclidean and integer grid point sets

New combinatorial properties identified

Abstract

There are many papers studying properties of point sets in the Euclidean space $\mathbb{E}^m$ or on integer grids $\mathbb{Z}^m$, with pairwise integral or rational distances. In this article we consider the distances or coordinates of the point sets which instead of being integers are elements of $\mathbb{Z} / \mathbb{Z}n$, and study the properties of the resulting combinatorial structures.