Paradoxical decompositions of finite-dimensional non-Archimedean normed spaces

Kamil Orzechowski

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arXiv · 2506.01528

Paradoxical decompositions of finite-dimensional non-Archimedean normed spaces

Kamil Orzechowski

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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.

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Kamil Orzechowski. 2025-06-02. Paradoxical decompositions of finite-dimensional non-Archimedean normed spaces. https://arxiv.org/abs/2506.01528

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