JournalsjemsVol. 24, No. 8pp. 2751–2773

Measurable Hall’s theorem for actions of abelian groups

  • Tomasz Cieśla

    McGill University, Montreal, Canada
  • Marcin Sabok

    McGill University, Montreal, Canada; Polish Academy of Sciences, Warszawa, Poland
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Abstract

We prove a measurable version of the Hall marriage theorem for actions of finitely generated abelian groups. In particular, it implies that for free measure-preserving actions of such groups and measurable sets which are suitably equidistributed with respect to the action, if they are are equidecomposable, then they are equidecomposable using measurable pieces. The latter generalizes a recent result of Grabowski, Máthé and Pikhurko on the measurable circle squaring and confirms a special case of a conjecture of Gardner.

Cite this article

Tomasz Cieśla, Marcin Sabok, Measurable Hall’s theorem for actions of abelian groups. J. Eur. Math. Soc. 24 (2022), no. 8, pp. 2751–2773

DOI 10.4171/JEMS/1164