Cubic sublattices

TL;DR

This paper proves that any integer vector with squared length divisible by a perfect square is contained in a cubic sublattice of the integer lattice, using an elementary approach based on cross products, improving previous quaternion-based results.

Contribution

It provides an elementary proof characterizing cubic sublattices containing specific vectors, improving upon prior quaternion-based methods.

Findings

Existence of cubic sublattices containing vectors with squared length divisible by d^2

Elementary proof using cross product techniques

Characterization of cubic sublattices in ^3

Abstract

A sublattice of the three-dimensional integer lattice $\mathbb Z^3$ is called cubic sublattice if there exists a basis of the sublattice whose elements are pairwise orthogonal and of equal lengths. We show that for an integer vector $\mathbf v\in\mathbb Z^3$ whose squared length is divisible by $d^2$, there exists a cubic sublattice containing $\mathbf v$ with edge length $d$. This improves one of the main result of a paper [arxiv:0806.3943] of Goswick et al., where similar theorem was proved by using the decomposition theory of Hurwitz integral quaternions. We give an elementary proof heavily using cross product. This method allows us to characterize the cubic sublattices.