Abstract
Laczkovich proved that if bounded subsets $A$ and $B$ of $\mathbb{R}^k$ have the same nonzero Lebesgue measure and the upper box dimension of the boundary of each set is less than $k$, then there is a partition of $A$ into finitely many parts that can be translated to form a partition of $B$. Here we show that it can be additionally required that each part is both Baire and Lebesgue measurable. As special cases, this gives measurable and translation-only versions of Tarski’s circle squaring and Hilbert’s third problem.
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title = {O r{ó}wnowaz{ń}os{ć}i wielok{\c a}t{ó}w (in {P}olish, with {F}rench summary)},
pages = {54},
year = {1924},
journal = {Przeglad Matematyczno-Fizyczny},
} -
[tarski:25] A. Tarski, "Problème 38," Fund. Math., vol. 7, p. 381, 1925.
@ARTICLE{tarski:25, volume = {7},
author = {Tarski, A.},
title = {Probl{è}me 38},
pages = {381},
year = {1925},
journal = {Fund. Math.},
} -
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@ARTICLE{timar:04, mrkey = {2081459},
issn = {1083-589X},
author = {Tim{á}r, {Á}d{á}m},
mrclass = {60G55 (05C70 05C80 60K35)},
doi = {10.1214/ECP.v9-1073},
journal = {Electron. Comm. Probab.},
zblnumber = {1060.60050},
volume = {9},
mrnumber = {2081459},
fjournal = {Electronic Communications in Probability},
mrreviewer = {Anton Wakolbinger},
title = {Tree and grid factors for general point processes},
year = {2004},
pages = {53--59},
} -
[timar:09:arxiv] Ádám. Timár, Invariant matchings of exponential tail on coin flips in $Z^d$, 2009.
@MISC{timar:09:arxiv,
author = {Tim{á}r, {Á}d{á}m},
arxiv = {0909.1090},
title = {Invariant matchings of exponential tail on coin flips in {$Z^d$}},
year = {2009},
} -
[tomkowicz+wagon:btp] G. Tomkowicz and S. Wagon, The Banach-Tarski Paradox, 2nd Edition, Cambridge: Cambridge Univ. Press, 2016.
@BOOK{tomkowicz+wagon:btp, zblnumber = {06595361},
author = {Tomkowicz, G. and Wagon, S.},
titlenote = {with a foreword by {J}an {M}ycielski},
title = {The {B}anach-{T}arski Paradox, 2nd Edition},
address = {Cambridge},
publisher = {Cambridge Univ. Press},
year = {2016},
} -
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@ARTICLE{wagon:81, mrkey = {0642135},
number = {4},
issn = {0343-6993},
author = {Wagon, Stanley},
mrclass = {51-02 (03E20 28A12 51M30)},
doi = {10.1007/BF03022979},
journal = {Math. Intelligencer},
zblnumber = {0479.01009},
volume = {3},
mrnumber = {0642135},
fjournal = {The Mathematical Intelligencer},
mrreviewer = {J. Strommer},
coden = {MAINDC},
title = {Circle-squaring in the twentieth century},
year = {1980/81},
pages = {176--181},
} -
[wagon:btp] S. Wagon, The Banach-Tarski paradox, Cambridge: Cambridge Univ. Press, 1985.
@BOOK{wagon:btp, mrkey = {0803509},
number = {24},
author = {Wagon, Stanley},
mrclass = {04A25 (03E25 04-02 28-02 43A05)},
series = {Encyclopedia Math. Appl.},
isbn = {0-521-30244-7},
address = {Cambridge},
publisher = {Cambridge Univ. Press},
doi = {10.1017/CBO9780511609596},
zblnumber = {0569.43001},
mrnumber = {0803509},
titlenote = {with a foreword by Jan Mycielski},
mrreviewer = {W{\l}odzimierz Bzyl},
title = {The {B}anach-{T}arski paradox},
year = {1985},
pages = {xvi+251},
}
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