A classical result by Pál Turán, estimates the global behavior
of an exponential polynomial on an interval by its supremum on any
arbitrary subinterval. In this talk we discuss Nazarov's extension of this
``global to local reduction'' to arbitrary Borel sets of positive Lebesgue
measure.

Using the idea of "acceleration", which is first introduced by Avila, I will construct a
cerntain class of analytic quasiperiodic Szego cocycles with uniformly positive LE.

We will present the combinatorial principles ITP and ISP, discuss how they strengthen the
tree property, and talk about their relative consistency