The spectrum of the weakly coupled Fibonacci Hamiltonian

We consider the spectrum of the Fibonacci Hamiltonian for small values of thecoupling constant. It is known that this set is a Cantor set of zero Lebesguemeasure. Here we study the limit, as the value of the coupling constantapproaches zero, of its thickness and its Hausdorff dimension. We announce thefollowing results and explain some key ideas that go into their proofs. Thethickness tends to infinity and, consequently, the Hausdorff dimension of thespectrum tends to one. Moreover, the length of every gap tends to zerolinearly. Finally, for sufficiently small coupling, the sum of the spectrumwith itself is an interval. This last result provides a rigorous explanation ofa phenomenon for the Fibonacci square lattice discovered numerically byEven-Dar Mandel and Lifshitz.