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Answer to a question of Kolmogorov


Authors: Richárd Balka, Márton Elekes and András Máthé
Journal: Proc. Amer. Math. Soc. 143 (2015), 2085-2089
MSC (2010): Primary 28A75, 26A16
DOI: https://doi.org/10.1090/S0002-9939-2014-12388-4
Published electronically: December 15, 2014
MathSciNet review: 3314117
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Abstract: More than 80 years ago Kolmogorov asked the following question. Let $ E\subseteq \mathbb{R}^{2}$ be a measurable set with $ \lambda ^{2}(E)<\infty $, where $ \lambda ^2$ denotes the two-dimensional Lebesgue measure. Does there exist for every $ \varepsilon >0$ a contraction $ f\colon E\to \mathbb{R}^{2}$ such that $ \lambda ^{2}(f(E))\geq \lambda ^{2}(E)-\varepsilon $ and $ f(E)$ is a polygon? We answer this question in the negative by constructing a bounded, simply connected open counterexample. Our construction can easily be modified to yield an analogous result in higher dimensions.


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Additional Information

Richárd Balka
Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, 1364 Budapest, Hungary
Address at time of publication: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195-4350
Email: balka.richard@renyi.mta.hu

Márton Elekes
Affiliation: Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, 1364 Budapest, Hungary – and – Institute of Mathematics, Eötvös Loránd University, Pázmány Péter s. 1/c, 1117 Budapest, Hungary
Email: elekes.marton@renyi.mta.hu

András Máthé
Affiliation: Mathematics Institute, University of Warwick, Coventry, CV4 7AL, United Kingdom
Email: A.Mathe@warwick.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-2014-12388-4
Keywords: Lebesgue measure, Lipschitz map, contraction, polygon
Received by editor(s): February 1, 2013
Received by editor(s) in revised form: October 16, 2013
Published electronically: December 15, 2014
Additional Notes: The authors gratefully acknowledge the support of the Hungarian Scientific Research Fund grants no. 72655 and 104178
The second author was supported by the Hungarian Scientific Research Fund grant no. 83726
The third author was supported by the Leverhulme Trust
Communicated by: Tatiana Toro
Article copyright: © Copyright 2014 American Mathematical Society