Undecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile

We construct an example of a group [Formula: see text] for a finite abelian group [Formula: see text] , a subset E of [Formula: see text] , and two finite subsets [Formula: see text] of G, such that it is undecidable in ZFC whether [Formula: see text] can be tiled by translations of [Formula: see text] . In particular, this implies that this tiling problem is aperiodic, in the sense that (in the standard universe of ZFC) there exist translational tilings of E by the tiles [Formula: see text] , but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in [Formula: see text] ). A similar construction also applies for [Formula: see text] for sufficiently large d. If one allows the group [Formula: see text] to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile F. The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles.