Abstract
Gardner conjectured that if two bounded measurable sets A, B ⊂ ℝn are equidecomposable by a set of isometries Γ generating an amenable group then A and B admit a measurable equidecomposition by all isometries. Cieśla and Sabok asked if there is a measurable equidecomposition using isometries only in the group generated by Γ. We answer this question negatively.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Banach and A. Tarski: Sur la décomposition des ensembles de points en parties respectivement congruentes, Fund. math 6.1 (1924), 244–277.
T. Cieśla and M. Sabok: Measurable Hall’s theorem for actions of Abelian groups, arXiv:1903.02987 (2019).
R. J. Gardner: Measure theory and some problems in geometry, Atti Sem. Mat. Fis. Univ. Modena 39 (1991), 51–72.
L. Grabowski, A. Máthá and O. Pikhurko: Measurable circle squaring, Annals of Mathematics (2017), 671–710.
A. S. Kechris and A. S. Marks: Descriptive graph combinatorics, 2015, preprint.
M. Laczkovich: Closed sets without measurable matching, Proceedings of the American Mathematical Society 103.3 (1988), 894–896.
M. Laczkovich: Equidecomposability and discrepancy; a solution of Tarski’s circle-squaring problem, Journal für die reine und angewandte Mathematik 404 (1990), 77–117.
A. S. Marks and S. T. Unger: Borel circle squaring, Annals of Mathematics 186 (2017), 581–605.
A. Máthé: Measurable equidecompositions, Proceedings of the International Congress of Mathematicians, 2, 2018.
J. Mycielski: Finitely additive invariant measures. I, in: Colloquium Mathematicum, Vol. 42, No. 1, (1979) 309–318, Institute of Mathematics Polish Academy of Sciences.
J. von Neumann: Zur allgemeinen Theorie des Masses, Fundamenta Mathematicae 13.1 (1929), 73–116.
A. Tarski: Probléme 38, Fundamenta Mathematicae 7 (1925), 381.
G. Tomkowicz and S. Wagon: The Banach-Tarski Paradox, Vol. 163, Cambridge University Press, 2016.
Acknowledgement
The author thanks to the anonymous referees for their remarks.
Author information
Authors and Affiliations
Corresponding author
Additional information
The author’s work on the project leading to this application has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 741420), from the UNKP-20-5 New National Excellence Program of the Ministry of Innovation and Technology from the source of the National Research, Development and Innovation Fund and from the Já;nos Bolyai Scholarship of the Hungarian Academy of Sciences.
