We develop some non-perturbative methods for studying the IDS in almost
Mathieu and related models. Assuming positive Lyapunov exponents, and assuming
that the potential function is a trigonometric polynomial of degree k, we show
that the Holder exponent of the IDS is almost 1/2k. We also show that this is
stable under small perturbations of the potential (e.g., potentials which are
close to that of almost Mathieu again give rise to almost 1/2 Holder continuous
IDS). Moreover, off a set of Hausdorff dimension zero the IDS is Lipschitz. We
further deduce from these properties that the IDS is absolutely continuous for
almost every shift angle. The proof is based on large deviation theorems for
the entries of the transfer matrices in combination with the avalanche
principle for non-unimodular matrices. These two are combined to yield a
multiscale approach to counting zeros of determinants which fall into small
disks. In addition, we develop a new mechanism of eliminating resonant phases,
and use it to obtain lower bounds on the gap between Dirichlet eigenvalues. The
latter also uses the exponential localization of the eigenfunctions.