Paradoxical decompositions of finite-dimensional non-Archimedean normed spaces

TL;DR

This paper demonstrates that finite-dimensional non-Archimedean normed spaces over certain fields can be decomposed paradoxically into four parts using affine isometries, extending the concept of Banach–Tarski paradox to non-Archimedean settings.

Contribution

It establishes the existence of paradoxical decompositions in finite-dimensional non-Archimedean normed spaces, a novel extension of classical paradoxical decompositions.

Findings

Paradoxical decompositions exist for spaces over non-Archimedean fields.

Four-piece paradoxical decompositions are possible using affine isometries.

Results extend Banach–Tarski paradox to non-Archimedean normed spaces.

Abstract

We show that any normed space $(K^n,\|\cdot\|)$, $n\ge 2$, 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 $\|\cdot\|$ is equivalent to the maximum norm. It follows that any finite-dimensional normed space $(X,\|\cdot\|)$ 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.