A trichotomy for a class of equivalence relations

TL;DR

This paper establishes a trichotomy for a class of Borel equivalence relations defined via summability conditions, classifying them into three distinct complexity categories, and characterizes when such relations are $ ext{ell}_p$-like.

Contribution

It proves a trichotomy theorem for Borel equivalence relations arising from summability conditions and characterizes their structure as $ ext{ell}_p$-like relations.

Findings

Either $ ext{R}^ ext{N}/ ext{ell}_1$ is Borel reducible to $E$, or $E_1$ is reducible to $E$, or $E$ is reducible to $E_0$.

Characterizes when these relations are $ ext{ell}_p$-like equivalence relations.

Provides a criterion for when such relations are equivalence relations.

Abstract

Let $X_n, n\in\Bbb N$ be a sequence of non-empty sets, $\psi_n:X_n^2\to\Bbb R^+$. We consider the relation $E((X_n,\psi_n)_{n\in\Bbb N})$ on $\prod_{n\in\Bbb N}X_n$ by $(x,y)\in E((X_n,\psi_n)_{n\in\Bbb N})\Leftrightarrow\sum_{n\in\Bbb N}\psi_n(x(n),y(n))<+\infty$. If $E((X_n,\psi_n)_{n\in\Bbb N})$ is a Borel equivalence relation, we show a trichotomy that either $\Bbb R^\Bbb N/\ell_1\le_B E$, $E_1\le_B E$, or $E\le_B E_0$. We also prove that, for a rather general case, $E((X_n,\psi_n)_{n\in\Bbb N})$ is an equivalence relation iff it is an $\ell_p$-like equivalence relation.