A counterexample to the periodic tiling conjecture

The periodic tiling conjecture asserts that any finite subset of a lattice Zᵈ 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 Rᵈ. In fact, we also obtain a counterexample in a group of the form Z² × 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.