  We consider the spectrum of the Fibonacci Hamiltonian for small values of the
coupling constant. It is known that this set is a Cantor set of zero Lebesgue
measure. Here we study the limit, as the value of the coupling constant
approaches zero, of its thickness and its Hausdorff dimension. We prove that
the thickness tends to infinity and, consequently, the Hausdorff dimension of
the spectrum tends to one. We also show that at small coupling, all gaps
allowed by the gap labeling theorem are open and the length of every gap tends
to zero linearly. Moreover, for sufficiently small coupling, the sum of the
spectrum with itself is an interval. This last result provides a rigorous
explanation of a phenomenon for the Fibonacci square lattice discovered
numerically by Even-Dar Mandel and Lifshitz. Finally, we provide explicit upper
and lower bounds for the solutions to the difference equation and use them to
study the spectral measures and the transport exponents.
