The Banach-Tarski paradox for some subsets of finite-dimensional normed spaces over non-Archimedean valued fields

2024/02/22 by Kamil Orzechowski, Orzechowski, Kamil · 1 citation
Mathematics · #03E25 #05A18 #12J25 #20E05 #20H20 #26E30 #46B04 #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #Group Theory (math.GR) #Mathematical and Theoretical Analysis #Primary 47S10 #Representation Theory (math.RT) #Secondary 46S10 #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2402.14772

openalex publication_date 2024/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show some results related to the classical Banach-Tarski paradox in the setting of finite-dimensional normed spaces over a non-Archimedean valued field K. For instance, all balls and spheres in Kⁿ, and the whole space Kⁿ (for n≥ 2) are paradoxical with respect to certain groups of isometries of Kⁿ. If K is locally compact (e.g., K is the field ℚ_p of p-adic numbers for any prime number p), any two bounded subsets of Kⁿ with nonempty interiors are equidecomposable (and paradoxical) with respect to a certain group of isometries of Kⁿ (for n≥ 2).

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• Paradoxical decompositions of finite-dimensional non-Archimedean normed spaces

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