The Banach-Tarski paradox in complete discretely valued fields
Kamil Orzechowski
TL;DR
This work extends the Banach–Tarski paradox to one-dimensional non-Archimedean settings by proving that complete discretely valued fields $\mathbb{K}$, along with all balls and spheres within them, admit paradoxical decompositions under the isometry group $\mathrm{Iso}(\mathbb{K})$. The authors reduce the problem to the affine-isometry action on $\mathbb{K}^2$ with the $\| frac{\cdot}{\infty}\|$-norm via a Laurent-series based map $f:\mathbb{K}\to\mathbb{K}^2$, conjugate suitable affine isometries to isometries of $\mathbb{K}$, and leverage known two-dimensional results to obtain four-piece paradoxes for $\mathbb{K}$ (and four-piece decompositions for balls and spheres). For separable fields, they obtain six-piece decompositions with the Baire property, while locally compact fields (e.g., $\mathbb{Q}_p$) enjoy strong equidecomposability: any two bounded sets with nonempty interiors are $\mathrm{Iso}(\mathbb{K})$-equidecomposable (and Baire-equidecomposable when measurable). The results complete the non-Archimedean paradoxical-decomposition picture in dimension one, complementing existing higher-dimensional work and providing explicit constructions in the $p$-adic setting. The methods combine a Laurent-series representation, transfer through a conjugation map, and modern equidecomposability criteria (GrMaPikhurko, DF/MU).
Abstract
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.