Undecidability of Translational Tiling of the 4-dimensional Space with a Set of 4 Polyhypercubes

Recently, Greenfeld and Tao disproof the conjecture that translational tilings of a single tile can always be periodic Ann. Math. 200 (2024), 301-363. In another paper to appear in J. Eur. Math. Soc. , they also show that if the dimension n is part of the input, the translational tiling for subsets of Zⁿ with one tile is undecidable. These two results are very strong pieces of evidence for the conjecture that translational tiling of Zⁿ with a monotile is undecidable, for some fixed n. This paper shows that translational tiling of the 3-dimensional space with a set of 5 polycubes is undecidable. By introducing a technique that lifts a set of polycubes and its tiling from 3-dimensional space to 4-dimensional space, we manage to show that translational tiling of the 4-dimensional space with a set of 4 tiles is undecidable. This is a step towards the attempt to settle the conjecture of the undecidability of translational tiling of the n-dimensional space with a monotile, for some fixed n.