The spectrum of the weakly coupled Fibonacci Hamiltonian

We consider the spectrum of the Fibonacci Hamiltonian for smallvalues of the coupling constant. It is known that this set is aCantor set of zero Lebesgue measure. Here we study the limit, as thevalue of the coupling constant approaches zero, of its thickness andits Hausdorff dimension. We announce the following results andexplain some key ideas that go into their proofs. The thicknesstends to infinity and, consequently, the Hausdorff dimension of thespectrum tends to one. Moreover, the length of every gap tends tozero linearly. Finally, for sufficiently small coupling, the sum ofthe spectrum with itself is an interval. This last result provides arigorous explanation of a phenomenon for the Fibonacci squarelattice discovered numerically by Even-Dar Mandel and Lifshitz.