Rachel Greenfeld and I have just uploaded to the arXiv our paper “Undecidability of translational monotilings“. This is a sequel to our previous paper in which we constructed a translational monotiling of a high-dimensional lattice (thus the monotile is a finite set and the translates , of partition ) which was aperiodic (there is no way to “repair” this tiling into a periodic tiling , in which is now periodic with respect to a finite index subgroup of ). This disproved the periodic tiling conjecture of Stein, Grunbaum-Shephard and Lagarias-Wang, which asserted that such aperiodic translational monotilings do not exist. (Compare with the “hat monotile“, which is a recently discovered aperiodic isometric monotile for of , where one is now allowed to use rotations and reflections as well as translations, or the even more recent “spectre monotile“, which is similar except that no reflections are needed.)
One of the motivations of this conjecture was the observation of Hao Wang that if the periodic tiling conjecture were true, then the translational monotiling problem is (algorithmically) decidable: there is a Turing machine which, when given a dimension and a finite subset of , can determine in finite time whether can tile . This is because if a periodic tiling exists, it can be found by computer search; and if no tiling exists at all, then (by the compactness theorem) there exists some finite subset of that cannot be covered by disjoint translates of , and this can also be discovered by computer search. The periodic tiling conjecture asserts that these are the only two possible scenarios, thus giving the decidability.