Computing (or not) Quasi-Periodicity Functions of Tilings

Abstract. 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 dom...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Alexis Ballier, Emmanuel Jeandel
• Keywords: CORR 2010, Education, Quasi-periodic Tilings, Quasi-periodicity Function, Quasiperiodic Tiling