Dot-product sets and simplices over finite rings

TL;DR

This paper investigates the properties of dot-product sets and simplices in vector spaces over finite rings, establishing conditions under which these sets are large or dense, thus advancing understanding of geometric configurations in algebraic structures.

Contribution

It introduces new results on the size and density of dot-product sets and simplices over finite rings, extending geometric combinatorics to algebraic settings.

Findings

Dot-product sets cover the entire ring when E is sufficiently large.

In higher dimensions, simplices determined by E have positive density.

Set of dot-product simplices also has positive density under size conditions.

Abstract

In this paper, we study dot-product sets and $k$-simplices in vector spaces over finite rings. We show that if $E$ is sufficiently large then the dot-product set of $E$ covers the whole ring. In higher dimensional cases, if $E$ is sufficiently large then the set of simplices and the set of dot-product simplices determined by $E$, up to congurence, have positive densities.