Undecidability of Translational Tiling of the 3-dimensional Space with a Set of 6 Polycubes

This paper focuses on the undecidability of translational tiling of $n$-dimensional space ${\mathbb{Z}}^n$ with a set of $k$ tiles. It is known that tiling ${\mathbb{Z}}^2$ with translated copies with a set of $8$ tiles is undecidable. Greenfeld and Tao gave strong evidence in a series of works that for sufficiently large dimension $n$, the translational tiling problem for ${\mathbb{Z}}^n$ might be undecidable for just one tile. This paper shows the undecidability of translational tiling of ${\mathbb{Z}}^3$ with a set of $6$ tiles.