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

Kamil Orzechowski

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 $K^n$, and the whole space $K^n$ (for $n\ge 2$) are paradoxical with respect to certain groups of isometries of $K^n$. If $K$ is locally compact (e.g., $K$ is the field $\mathbb{Q}_p$ of $p$-adic numbers for any prime number $p$), any two bounded subsets of $K^n$ with nonempty interiors are equidecomposable (and paradoxical) with respect to a certain group of isometries of $K^n$ (for $n\ge 2$).

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

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 . For instance, all balls and spheres in , and the whole space (for ) are paradoxical with respect to certain groups of isometries of . If is locally compact (e.g., is the field of -adic numbers for any prime number ), any two bounded subsets of with nonempty interiors are equidecomposable (and paradoxical) with respect to a certain group of isometries of (for ).
Paper Structure (10 sections, 69 equations)

This paper contains 10 sections, 69 equations.

Table of Contents

  1. Introduction
  2. Non-Archimedean valued fields and normed spaces over them
  3. Preliminaries
  4. Groups of linear and affine isometries of normed spaces $(K^n, \lVert {\mspace{1mu}\cdot\mspace{1mu}} \rVert )$
  5. Paradoxical sets with respect to group actions
  6. Amenability and nonexistence of paradoxical decompositions
  7. Some free subgroups of special linear groups $\mathop{\mathrm{SL}}\nolimits(n,D)$ and special affine groups $\mathop{\mathrm{SA}}\nolimits(n,D)$
  8. The case when the characteristic of $K$ is zero
  9. The case when $K$ is a transcendental extension of its prime field
  10. The Banach--Tarski paradox for subsets of $K^n$

Theorems & Definitions (26)

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