On the Non-existence of Lattice Tilings by Quasi-crosses

TL;DR

This paper establishes necessary conditions and proves the non-existence of lattice tilings of Euclidean space by certain quasi-cross shapes, narrowing down the remaining unclassified cases for specific quasi-cross parameters.

Contribution

It provides new non-existence results for lattice tilings by quasi-crosses, especially focusing on the smallest unclassified shapes, and reduces the open cases for these tilings.

Findings

No lattice tilings for (3,1,n)-quasi-crosses in most dimensions up to 250.

No lattice tilings for (3,2,n)-quasi-crosses in most dimensions up to 250.

Identifies only a few remaining unclassified cases for these shapes.

Abstract

We study necessary conditions for the existence of lattice tilings of $\R^n$ by quasi-crosses. We prove non-existence results, and focus in particular on the two smallest unclassified shapes, the $(3,1,n)$-quasi-cross and the $(3,2,n)$-quasi-cross. We show that for dimensions $n\leq 250$, apart from the known constructions, there are no lattice tilings of $\R^n$ by $(3,1,n)$-quasi-crosses except for ten remaining cases, and no lattice tilings of $\R^n$ by $(3,2,n)$-quasi-crosses except for eleven remaining cases.