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Ponzi schemes on coarse spaces with uniform measure

Shunsuke Miyauchi

TL;DR

This work extends the Block–Weinberger and Roe framework by introducing a uniform measure $\mu$ on measurable coarse spaces and defining $\mu$-PS, a Ponzi-scheme analogue that leverages measure-theoretic tools. It proves a key equivalence: a $\mu$-PS together with a quasi lattice implies the existence of a Ponzi scheme, and conversely a Ponzi scheme with a $\mu$-constant controlled set yields a $\mu$-PS, with corollaries linking the non-amenability of a group acting properly and cocompactly to the presence of a $\mu$-PS on the induced coarse space. The paper also provides a concrete $\mu$-PS example on the hyperbolic disk and investigates the coarse invariance of $\mu$-PS under measure-appropriate coarse maps, establishing a framework in which amenability phenomena can be analyzed via measure-based Ponzi-type obstructions. Overall, it merges coarse geometry with measure theory to yield new tools for detecting amenability and understanding coarse dynamical properties.

Abstract

Ponzi schemes, defined by Block-Weinberger(1992) and Roe(2003), give a characterization of amenability from the viewpoint of coarse geometry. We consider measures in coarse spaces, and propose a reformulation of Ponzi schemes with measures.

Ponzi schemes on coarse spaces with uniform measure

TL;DR

This work extends the Block–Weinberger and Roe framework by introducing a uniform measure on measurable coarse spaces and defining -PS, a Ponzi-scheme analogue that leverages measure-theoretic tools. It proves a key equivalence: a -PS together with a quasi lattice implies the existence of a Ponzi scheme, and conversely a Ponzi scheme with a -constant controlled set yields a -PS, with corollaries linking the non-amenability of a group acting properly and cocompactly to the presence of a -PS on the induced coarse space. The paper also provides a concrete -PS example on the hyperbolic disk and investigates the coarse invariance of -PS under measure-appropriate coarse maps, establishing a framework in which amenability phenomena can be analyzed via measure-based Ponzi-type obstructions. Overall, it merges coarse geometry with measure theory to yield new tools for detecting amenability and understanding coarse dynamical properties.

Abstract

Ponzi schemes, defined by Block-Weinberger(1992) and Roe(2003), give a characterization of amenability from the viewpoint of coarse geometry. We consider measures in coarse spaces, and propose a reformulation of Ponzi schemes with measures.
Paper Structure (13 sections, 9 theorems, 34 equations, 2 figures)

This paper contains 13 sections, 9 theorems, 34 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Notations
  3. Uniform measure
  4. Measurablity of controlled sets
  5. Uniform measure
  6. Examples of measurable coarse structures and uniform measures
  7. An analogue of Ponzi scheme for coarse spaces with uniform measures
  8. Definition of $\mu$-PS and Ponzi scheme
  9. Review of the notions in the definition of Ponzi scheme
  10. The notions in the definition of $\mu$-PS
  11. Some remarks on Ponzi schemes and $\mu$-PS
  12. An example of $\mu$-PS
  13. Coarse property of $\mu$-PS

Key Result

theorem 1.1 thmcountertheorem

(See Theorem thm:muPS) Let $(X,\mathcal{E})$ be a measurable coarse space, $\mu$ be a uniform measure, and the measure space $(X,\mu)$ be $\sigma$-finite.

Figures (2)

  • Figure 1: Case 1. in the proof of Example \ref{['ex:muPS']}.
  • Figure 2: Case 2. in the proof of Example \ref{['ex:muPS']}.

Theorems & Definitions (31)

  • theorem 1.1 thmcountertheorem
  • corollary 1.1.1
  • definition 1.2.1
  • definition 1.2.2
  • definition 1.2.3
  • definition 1.2.4
  • definition 1.3.1
  • definition 1.3.2
  • proposition 1.3.1
  • definition 1.3.3
  • ...and 21 more