Note on distortion and Bourgain $\ell_1$ index

TL;DR

This paper investigates the relationship between measures of proximity to  and distortability in Banach spaces, establishing that spaces with high Bourgain  indices contain either distortable or ^-asymptotic subspaces.

Contribution

It proves that Banach spaces with all subspaces having Bourgain  index above a certain threshold necessarily contain specific types of subspaces, linking index bounds to structural properties.

Findings

Spaces with Bourgain  index >  contain distortable subspaces

Such spaces also contain ^-asymptotic subspaces

Main result connects index bounds to subspace structure

Abstract

The 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 $\omega^\alpha$, $\alpha<\omega_1$, contains either an arbitrary distortable subspace or an $\ell_1^\alpha$-asymptotic subspace.