Measurable Tilings by Abelian Group Actions

Let be a measure space with a measure-preserving action of an abelian group. We consider the problem of understanding the structure of measurable tilings of by a measurable tile translated by a finite set of shifts, thus the translates, partition up to null sets. Adapting arguments from previous literature, we establish a "dilation lemma"that asserts, roughly speaking, that implies for a large family of integer dilations, and use this to establish a structure theorem for such tilings analogous to that established recently by the second and fourth authors. As applications of this theorem, we completely classify those random tilings of finitely generated abelian groups that are "factors of iid", and show that measurable tilings of a torus can always be continuously (in fact linearly) deformed into a tiling with rational shifts, with particularly strong results in the low-dimensional cases (in particular resolving a conjecture of Conley, the first author, and Pikhurko in the case).