The Banach–Tarski paradox in complete discretely valued fields

We prove some results related to the classical BanachTarski paradox in the setting of a eld K that is complete with respect to a discrete non-Archimedean valuation (e.g., when K is the eld Q p of p-adic numbers for a prime p).Namely, the eld K, as well as all balls and spheres in K, admit a paradoxical decomposition with respect to the isometry group of K.Such decompositions can be realized using pieces with the Baire property if K is separable.Under the additional assumption of local compactness of K (e.g., when K = Q p ), any two bounded subsets of K with nonempty interiors are equidecomposable with respect to the isometry group of 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 K with respect to groups of ane isometries.