Paradoxical decompositions of finite-dimensional non-Archimedean normed spaces
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909632546471936 |
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| author | Orzechowski, Kamil |
| author_facet | Orzechowski, Kamil |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_01528 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Paradoxical decompositions of finite-dimensional non-Archimedean normed spaces Orzechowski, Kamil Functional Analysis Group Theory Primary 47S10, Secondary 46S10, 12J25, 26E30, 20E05, 20G25, 20H05, 22F05 46B04, 05A18 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. |
| title | Paradoxical decompositions of finite-dimensional non-Archimedean normed spaces |
| topic | Functional Analysis Group Theory Primary 47S10, Secondary 46S10, 12J25, 26E30, 20E05, 20G25, 20H05, 22F05 46B04, 05A18 |
| url | https://arxiv.org/abs/2506.01528 |