  We consider one-dimensional difference Schroedinger equations on the discrete
line with a potential generated by evaluating a real-analytic potential
function V(x) on the one-dimensional torus along an orbit of the shift
x-->x+nw. If the Lyapunov exponent is positive for all energies and w, then the
integrated density of states is absolutely continuous for almost every w. In
this work we establish the formation of a dense set of gaps in the spectrum.
Our approach is based on an induction on scales argument, and is therefore both
constructive as well as quantitative. Resonances between eigenfunctions of one
scale lead to "pre-gaps" at a larger scale. To pass to actual gaps in the
spectrum, we show that these pre-gaps cannot be filled more than a finite (and
uniformly bounded) number of times. To accomplish this, we relate a pre-gap to
pairs of complex zeros of the Dirichlet determinants off the unit circle using
the techniques of an earlier paper by the authors. Amongst other things, we
establish in this work a non-perturbative version of the co-variant
parametrization of the eigenvalues and eigenfunctions via the phases in the
spirit of Sinai's (perturbative) description of the spectrum via his function
as well as an multi-scale/finite volume approach to Anderson localization in
this context. This allows us to relate the gaps in the spectrum with the graphs
of the eigenvalues parametrized by the phase. Our infinite volume theorems hold
for all Diophantine frequencies w up to a set of Hausdorff dimension zero. This
is in contrast to earlier technology which only yielded exceptional sets of
measure zero. Our only assumption on the potential apart from analyticity is
the positivity of the Lyapunov exponent.
