Computing (or not) Quasi-periodicity Functions of Tilings

We know that tilesets that can tile the plane always admit a quasiperiodic tiling [4, 8], yet they hold many uncomputable properties [3, 11, 21, 25]. The quasi-periodicity function is one way to measure the regularity of a quasiperiodic tiling. We prove that the tilings by a tileset that admits only quasiperiodic tilings have a recursively (and uniformly) bounded quasi-periodicity function. This corrects an error from [6, theorem 9] which stated the contrary. Instead we construct a tileset for which any quasi-periodic tiling has a quasi-periodicity function that cannot be recursively bounded. We provide such a construction for 1−dimensional effective subshifts and obtain as a corollary the result for tilings of the plane via recent links between these objects [1, 10]. Tilings of the discrete plane as studied nowadays have been introduced by Wang in order to study the decidability of a subclass of first order logic [26, 27, 5]. After Berger proved the undecidability of the domino problem [3], interest has grown for understanding how complex are these simply defined objects [11, 21, 9, 6]. Despite being able to have complex tilings, any tileset that can tile the plane admits a quasiperiodic tiling [4, 8]; roughly speaking, a quasi-periodic tiling is a tiling in which every finite pattern can be found in any sufficiently large part of the tiling. It is therefore natural to define the quasi-periodicity function of a quasi-periodic tiling: it associates to an integer n the minimal size in which we are certain to find any pattern of size n [8, 6]. This is one way to measure the complexity of a quasi-periodic tiling and, to some extent, of a tileset τ since τ must admit at least one quasi-periodic tiling. We start by proving in Section 2 that tilings by tilesets that admit only quasi-periodic tilings have a recursively (and uniformly) bounded quasi-periodicity function (Theorem 1.4). Remark that there exists non-trivial tilesets that admit only quasi-periodic tilings [23, 19, 22] and that the property of having only such tilings can be reduced to the domino problem [3, 23] and is thus undecidable. Both authors are partly supported by ANR-09-BLAN-0164. A. Ballier has been partly supported by the Academy of Finland project 131558. We thank Pierre Guillon for discussions that lead to the constructions provided in Section 3. 1Take a tileset τu that admits only one uniform tiling (and thus only quasi-periodic tilings), a tileset τf that admits non quasi-periodic tilings (e.g., a fullshift on {0, 1}) then it is clear that (τ × τf ) ∪ τu admits only quasi-periodic tilings if and only if τ does not tile the plane.