Embeddings between sequence variable Lebesgue spaces, strict and finitely strict singularity

In this paper, we provide necessary and sufficient conditions under which two sequence variable Lebesgue spaces $\ell_{p_n}$ and $\ell_{q_n}$ are equivalent and also describe conditions under which the natural embeddings $id:\ell_{p_n} \to \ell_{q_n}$ are strictly or finitely strictly singular. We also provide estimates for the Bernstein numbers of the natural embedding $id$ and show how they depend on the exponents $p_n$ and $q_n$.