This is off-topic: But powerful techniques to show that gap exist, can be found in the following papers:
Michael Goldstein, Wilhelm Schlag, On resonances and the formation of gaps in the spectrum of quasi-periodic Schroedinger equations
Artur Avila, Jairo Bochi, David Damanik, Opening Gaps in the Spectrum of Strictly Ergodic Schrödinger Operators
of course. The setting is somewhat different, since they consider discrete Schroedinger operators ...
