Paradoxical decompositions of finite-dimensional non-Archimedean normed spaces

Abstract

We show that any normed space $(K^n,\Vert \cdot \Vert )$, $n\ge 2$, over a field $K$ equipped with a nontrivial non-Archimedean valuation admits a paradoxical decomposition using four pieces with respect to the group of its affine isometries, provided that the norm $\Vert \cdot \Vert$ is equivalent to the maximum norm. It follows that any finite-dimensional normed space $(X,\Vert \cdot \Vert )$ with $\dim X\ge 2$ over a complete non-Archimedean nontrivially valued field $(K,|\cdot |)$ is paradoxical using four pieces with respect to the group of its affine isometries.

15 pages