  The thrust distribution in e+e- annihilation is calculated exploiting its
exponentiation property in the two-jet region t = 1-T << 1. We present a
general method (DGE) to calculate a large class of logarithmically enhanced
terms, using the dispersive approach in renormalon calculus. Dressed Gluon
Exponentiation is based on the fact that the exponentiation kernel is
associated primarily with a single gluon emission, and therefore the exponent
is naturally represented as an integral over the running coupling. Fixing the
definition of Lambda is enough to guarantee consistency with the exact exponent
to next-to-leading logarithmic accuracy. Renormalization scale dependence is
avoided by keeping all the logs. Sub-leading logs, that are usually neglected,
are factorially enhanced and are therefore important. Renormalization-group
invariance as well as infrared renormalon divergence are recovered in the sum
of all the logs. The logarithmically enhanced cross-section is evaluated by
Borel summation. Renormalon ambiguity is then used to study power corrections
in the peak region Qt \gsim Lambda, where the hierarchy between the renormalon
closest to the origin (~1/Qt) and others (~1/(Qt)^n) starts to break down. The
exponentiated power-corrections can be described by a shape-function, as
advocated by Korchemsky and Sterman. Our calculation suggests that the even
central moments of the shape-function are suppressed. Good fits are obtained
yielding alpha_s^{MSbar} (M_Z) = 0.110 \pm 0.001, with a theoretical
uncertainty of ~5%.
