Proximity to $\ell_1$ and Distortion in Asymptotic $\ell_1$ Spaces

For an asymptotic $\ell_1$ space $X$ with a basis $(x_i)$ certain asymptotic $\ell_1$ constants, $\delta_\alpha (X)$ are defined for $\alpha <\omega_1$. $\delta_\alpha (X)$ measures the equivalence between all normalized block bases $(y_i)_{i=1}^k$ of $(x_i)$ which are $S_\alpha$-admissible with respect to $(x_i)$ ($S_\alpha$ is the $\alpha^{th}$-Schreier class of sets) and the unit vector basis of $\ell_1^k$. This leads to the concept of the delta spectrum of $X$, $\Delta (X)$, which reflects the behavior of stabilized limits of $\delta_\alpha (X)$. The analogues of these constants under all renormings of $X$ are also defined and studied. We investigate $\Delta (X)$ both in general and for spaces of bounded distortion. We also prove several results on distorting the classical Tsirelson's space $T$ and its relatives.