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For a separable rearrangement invariant space $X$ on $(0,\infty)$ offundamental 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 basisvectors of $\ell^p$ ($c_0$ if $p=\infty$) correspond to pairwise disjoint andequimeasurable functions. This characterization hinges upon a description ofthe set of approximate eigenvalues of the doubling operator $x(t)\mapstox(t/2)$ in $X$. We prove that this set is surprisingly simple: depending on thevalues of some dilation indices of such a space, it is either an interval or aunion of two intervals. We apply these results to the Lorentz and Orliczspaces.

Symmetric finite representability of $\ell^p$-spaces in rearrangement invariant spaces on $(0,\infty)$