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

Abstract

For $f \colon [0,1] \rar \real^{+}$, consider the relation ${\mathbf{E}}_f$ on $[0,1{]}^{\omega }$ defined by $(x_n){\mathbf{E}}_f(y_n)\iff {\sum }_{n<\omega }f(|y_n-x_n|)<\infty .$ We study the Borel reducibility of Borel equivalence relations of the form ${\mathbf{E}}_f$. Our results indicate that for every $1\le p<q<\infty$, the order ${\le }_B$ 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.