2026/02/09 by Kamil Orzechowski
Mathematics · #math.FA #math.GR #math.LO
We prove some results related to the classical Banach--Tarski paradox in the setting of a field \mathbbK that is complete with respect to a discrete non-Archimedean valuation (e.g., when \mathbbK is the field ℚp of p-adic numbers for a prime p). Namely, the field \mathbbK, as well as all balls and spheres in \mathbbK, admit a paradoxical decomposition with respect to the isometry group of \mathbbK. Such decompositions can be realized using pieces with the Baire property if \mathbbK is separable. Under the additional assumption of local compactness of \mathbbK (e.g., when \mathbbK=ℚp), any two bounded subsets of \mathbbK with nonempty interiors are equidecomposable with respect to the isometry group of \mathbbK. 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 \mathbbK with respect to groups of affine isometries.