A counterexample to the periodic tiling conjecture

The periodic tiling conjecture asserts that any finite subset of a lattice Z^d that tiles that lattice by translations, in fact tiles periodically. In this work we disprove this conjecture for sufficiently large d, which also implies a disproof of the corresponding conjecture for Euclidean spaces R^d. In fact, we also obtain a counterexample in a group of the form Z² x G₀ for some finite abelian 2-group G₀. Our methods rely on encoding a "Sudoku puzzle" whose rows and other non-horizontal lines are constrained to lie in a certain class of "2-adically structured functions," in terms of certain functional equations that can be encoded in turn as a single tiling equation, and then demonstrating that solutions to this Sudoku puzzle exist, but are all non-periodic.