Measurable Tilings by Abelian Group Actions

Abstract Let $X$ be a measure space with measure-preserving action $(g,x) \mapsto g \cdot x$ of an abelian group $G$. We consider the problem understanding structure measurable tilings $F \odot A = X$ by tile $A \subset translated finite set G$ shifts, thus translates $f A$, \in F$ partition up to null sets. Adapting arguments from previous literature, we establish “dilation lemma” that asserts, roughly speaking, implies $F^{r} for large family integer dilations $r$, and use this theorem such analogous established recently second fourth authors. As applications theorem, completely classify those random finitely generated groups are “factors iid”, show torus ${\mathbb{T}}^{d}$ can always continuously (in fact linearly) deformed into tiling rational particularly strong results in low-dimensional cases $d=1,2$ particular resolving conjecture Conley, first author, Pikhurko $d=1$ case).