2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/156769The relation between different notions measuring proximity to $\ell_1$ and distortability of a Banach space is studied. The main result states that a Banach space, whose all subspaces have Bourgain $\ell_1$ index greater than $ω^α$, $α<ω_1$, contains either an arbitrary distortable subspace or an $\ell_1^α$-asymptotic subspace.14 pages, remarks and references addedFunctional Analysis46B20 (Primary); 46B03 (Secondary)text