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

We construct a unilateral lattice tiling of (Formula presented.) into hypercubes of two differnet side lengths p or q. This generalizes the Pythagorean tiling in (Formula presented.). We also show that this tiling is unique up to symmetries, which proves a variation of a conjecture by Bölcskei from 2001. For positive integers p and q, this tiling also provides a tiling of (Formula presented.).