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

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 $(\mathbb{Z}\mathrm{/}(p^n+q^n)\mathbb{Z}{)}^n$.