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The Banach–Tarski paradox in complete discretely valued fields
- Part of
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- Journal:
- Canadian Mathematical Bulletin , First View
- Published online by Cambridge University Press:
- 11 August 2026, pp. 1-14
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We prove some results related to the classical Banach–Tarski paradox in the setting of a field $\mathbb {K}$
that is complete with respect to a discrete non-Archimedean valuation (e.g., when $\mathbb {K}$
is the field $\mathbb {Q}_p$
of $p$
-adic numbers for a prime $p$
). Namely, the field $\mathbb {K}$
, as well as all balls and spheres in $\mathbb {K}$
, admit a paradoxical decomposition with respect to the isometry group of $\mathbb {K}$
. Such decompositions can be realized using pieces with the Baire property if $\mathbb {K}$
is separable. Under the additional assumption of local compactness of $\mathbb {K}$
(e.g., when $\mathbb {K}=\mathbb {Q}_p$
), any two bounded subsets of $\mathbb {K}$
with nonempty interiors are equidecomposable with respect to the isometry group of $\mathbb {K}$
. Our results complete the study of paradoxical decompositions in the non-Archimedean setting, addressing the one-dimensional case and building on earlier work for higher-dimensional normed spaces over $\mathbb {K}$
with respect to groups of affine isometries.
