Borel reducibility and finitely Holder(\alpha) embeddability

TL;DR

This paper explores the relationship between Borel reducibility of certain equivalence relations and finitely H"older() embeddability of pseudo-metric spaces, providing insights into their structural connections.

Contribution

It establishes a novel connection between Borel reducibility and finitely HF6lder() embeddability, advancing understanding of the complexity of equivalence relations.

Findings

Borel reducibility is closely related to finitely HF6lder() embeddability.

Provides criteria for Borel reducibility based on embeddability properties.

Enhances the theoretical framework linking equivalence relations and metric space embeddings.

Abstract

Let $(X_n,d_n),\,n\in\Bbb N$ be a sequence of pseudo-metric spaces, $p\ge 1$. For $x,y\in\prod_{n\in\Bbb N}X_n$, let $(x,y)\in E((X_n)_{n\in\Bbb N};p)\Leftrightarrow\sum_{n\in\Bbb N}d_n(x(n),y(n))^p<+\infty$. For Borel reducibility between equivalence relations $E((X_n)_{n\in\Bbb N};p)$, we show it is closely related to finitely H\"older($\alpha$) embeddability between pseudo-metric spaces.