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

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