The periodic tiling conjecture in dimension three

Theorem 1.1: there is a finite nonempty $T\subset\mathbb Z^3$ that tiles $\mathbb Z^3$ by translations, but no translation set of a tiling is invariant under a finite-index subgroup; and $T+[0,1]^3$ tiles $\mathbb R^3$ up to null sets, with no tiling whose translation set is invariant under a full-rank lattice, even allowing real translations. With Bhattacharya's planar theorem, three is the least lattice dimension where periodic tiling fails. The tile is not claimed to be connected, and the result concerns full periodicity only, not the absence of every individual period.

Theorem 1.1: there is a finite nonempty $T\subset\mathbb Z^3$ that tiles $\mathbb Z^3$ by translations, but no translation set of a tiling is invariant under a finite-index subgroup; and $T+[0,1]^3$ tiles $\mathbb R^3$ up to null sets, with no tiling whose translation set is invariant under a full-rank lattice, even allowing real translations. With Bhattacharya's planar theorem, three is the least lattice dimension where periodic tiling fails. The tile is not claimed to be connected, and the result concerns full periodicity only, not the absence of every individual period.