On $\ell^{p}$-like equivalence relations

TL;DR

This paper investigates the complexity of certain equivalence relations defined by summability conditions on sequences, revealing intricate ordering structures in their Borel reducibility hierarchy.

Contribution

It characterizes the Borel reducibility order of $E_f$ relations for $ ext{Id}^p$ functions, showing the order's complexity surpasses initial expectations.

Findings

The reducibility order between $E_{ ext{Id}^p}$ and $E_{ ext{Id}^q}$ is highly intricate.

Every linear order of continuum size can embed into the reducibility hierarchy.

The hierarchy exhibits more complexity than previously anticipated.

Abstract

For $f \colon [0,1] \rar \real^{+}$, consider the relation $\mathbf{E}_{f}$ on $[0,1]^{\omega}$ defined by $(x_{n}) \mathbf{E}_{f} (y_{n}) \Leftrightarrow \sum_{n < \omega} f(|y_{n} - x_{n}|) < \infty.$ We study the Borel reducibility of Borel equivalence relations of the form $\mathbf{E}_{f}$. Our results indicate that for every $1 \leq p < q < \infty$, the order $\leq_{B}$ of Borel reducibility on the set of equivalence relations $\{\bE \colon \bE_{\Id^{p}} \leq_{B} \bE \leq_{B} \bE_{\Id^{q}}\}$ is more complicated than expected, e.g. consistently every linear order of cardinality continuum embeds into it.