On ℓp-like Equivalence Relations

Abstract

For $f : [0,1] \to \mathbb{R}^{+}$, 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 $\mathbf{E}:\mathbf{E}_{Id^\mathcal{p}} \leq_B \mathbf{E} \leq_B \mathbf{E}_{Id^\mathcal{q}}$ is more complicated than expected, e.g.\ consistently every linear order of cardinality continuum embeds into it