  We know that tilesets that can tile the plane always admit a quasi-periodic
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
quasi-periodic tiling. We prove that the tilings by a tileset that admits only
quasi-periodic 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].
