Filling space with hypercubes of two sizes -- The pythagorean tiling in higher dimensions

TL;DR

This paper generalizes the Pythagorean tiling to higher dimensions by constructing a unique lattice tiling of space with hypercubes of two sizes, extending previous two-dimensional results and confirming a conjecture.

Contribution

It introduces a higher-dimensional hypercube tiling with two sizes, proving its uniqueness and connecting it to a modular tiling, thus extending the Pythagorean tiling concept.

Findings

Constructed a higher-dimensional hypercube tiling with two sizes.

Proved the tiling's uniqueness up to symmetries.

Connected the tiling to modular arithmetic for integer parameters.

Abstract

We construct a unilateral lattice tiling of $\mathbb{R}^n$ into hypercubes of two differnet side lengths $p$ or $q$. This generalizes the Pythagorean tiling in $\mathbb{R}^2$. We also show that this tiling is unique up to symmetries, which proves a variation of a conjecture by B\"olcskei from 2001. For positive integers $p$ and $q$ this tiling also provides a tiling of $(\mathbb{Z}/(p^n+q^n)\mathbb{Z})^n$.