Extensions of integral orthoregular sets and icubes

TL;DR

This paper investigates conditions under which integral vectors can be extended to orthogonal bases of equal length, providing necessary and sufficient conditions in dimensions 3 and 4.

Contribution

It offers new criteria for extending integral vectors to orthogonal bases of equal length, with proven sufficiency in specific low dimensions.

Findings

Necessary conditions for extension are identified.

Sufficient conditions are established in dimensions 3 and 4.

Results apply to rational and Gaussian integral vectors.

Abstract

In this paper, we study the question when a (rational or Gaussian) integral vector can be extended to an integral orthogonal basis consisting of vectors of equal length. We also study when a set of integral vectors has such an extension. Some necessary conditions are given which are proven to be sufficient in dimensions $3$ and $4$.