  We consider Frenkel-Kontorova models corresponding to 1 dimensional
quasicrystals.
  We present a KAM theory for quasi-periodic equilibria. The theorem presented
has an \emph{a-posteriori} format. We show that, given an approximate solution
of the equilibrium equation, which satisfies some appropriate non-degeneracy
conditions, then, there is a true solution nearby. This solution is locally
unique.
  Such a-posteriori theorems can be used to validate numerical computations and
also lead immediately to several consequences a) Existence to all orders of
perturbative expansion and their convergence b) Bootstrap for regularity c) An
efficient method to compute the breakdown of analyticity.
  Since the system does not admit an easy dynamical formulation, the method of
proof is based on developing several identities. These identities also lead to
very efficient algorithms.
