  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.
