Paradoxical decompositions of finite-dimensional non-Archimedean normed spaces

2025/06/02 by Orzechowski, Kamil
#05A18 #12J25 #20E05 #20G25 #20H05 #22F05 46B04 #26E30 #FOS: Mathematics #Functional Analysis (math.FA) #Group Theory (math.GR) #Primary 47S10 #Secondary 46S10

paper · doi:10.48550/arxiv.2506.01528

Abstract

We show that any normed space (Kⁿ,‖⋅‖), n≥ 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 ‖⋅‖ is equivalent to the maximum norm. It follows that any finite-dimensional normed space (X,‖⋅‖) with dimX≥ 2 over a complete non-Archimedean nontrivially valued field (K,|⋅|) is paradoxical using four pieces with respect to the group of its affine isometries.

Citations

• The Banach-Tarski paradox for some subsets of finite-dimensional normed spaces over non-Archimedean valued fields
• Uniform Kazhdan Constants and Paradoxes of the Affine Plane
• Sur la décomposition des ensembles de points en parties respectivement congruentes

Related

• The Banach-Tarski paradox for some subsets of finite-dimensional normed spaces over non-Archimedean valued fields
• The Banach-Tarski paradox in complete discretely valued fields
• On the concept of non-ultrametric non-Archimedean analysis
• A Mazur-Ulam theorem for Mappings of conservative distance in non-Archimedean n-normed spaces
• A non-Archimedean Arens--Eells isometric embedding theorem on valued fields