Measurable Tilings by Abelian Group Actions

Jan Grebík, Rachel Greenfeld, Václav Rozhoň, Terence Tao

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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).

Original languageEnglish
Pages (from-to)20211-20251
Number of pages41
JournalInternational Mathematics Research Notices
Issue number23
StatePublished - 1 Dec 2023
Externally publishedYes

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© 2023 The Author(s). Published by Oxford University Press. All rights reserved.


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