  For a separable rearrangement invariant space $X$ on $(0,\infty)$ of
fundamental type we identify the set of all $p\in [1,\infty]$ such that
$\ell^p$ is finitely represented in $X$ in such a way that the unit basis
vectors of $\ell^p$ ($c_0$ if $p=\infty$) correspond to pairwise disjoint and
equimeasurable functions. This characterization hinges upon a description of
the set of approximate eigenvalues of the doubling operator $x(t)\mapsto
x(t/2)$ in $X$. We prove that this set is surprisingly simple: depending on the
values of some dilation indices of such a space, it is either an interval or a
union of two intervals. We apply these results to the Lorentz and Orlicz
spaces.
