 
Summary: Distorting Mixed Tsirelson Spaces
G. Androulakis
E. Odell
Abstract: Any regular mixed Tsirelson space T(n, Sn)IN for which n
n 0, where = limn 1/n
n ,
is shown to be arbitrarily distortable. Certain asymptotic 1 constants for those and other mixed
Tsirelson spaces are calculated. Also a combinatorial result on the Schreier families (S)<1 is
proved and an application is given to show that for every Banach space X with a basis (ei), the
two spectrums (X) and (X, (ei)) coincide.
1 Introduction
A Banach space X with basis (ei) is asymptotic 1 if there exists > 0 such that for all n and block
bases (xi)n
1 of (ei)
n ,
n
i=1
xi
n
i=1
