The Banach-Tarski paradox in complete discretely valued fields

TL;DR

This paper extends the Banach-Tarski paradox to complete discretely valued fields like the p-adic numbers, showing paradoxical decompositions and equidecomposability of sets within this non-Archimedean framework.

Contribution

It proves paradoxical decompositions for fields with non-Archimedean valuations and establishes conditions for equidecomposability of bounded sets, completing the non-Archimedean case.

Findings

Fields and spheres admit paradoxical decompositions

Bounded sets with nonempty interior are equidecomposable

Results apply to p-adic numbers and similar fields

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.