A dense, positive-Hausdorff-dimension set of frequencies gives the multidimensional almost Mathieu operator a zero-measure Cantor spectrum, and critical almost Mathieu spectra can have arbitrarily small positive box dimension.
Critical almost Mathieu operator: hidden singularity, gap continuity, and the Hausdorff dimension of the spectrum
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abstract
We prove almost Lipshitz continuity of spectra of singular quasiperiodic Jacobi matrices and obtain a representation of the critical almost Mathieu family that has a singularity. This allows us to prove that the Hausdorff dimension of its spectrum is not larger than 1/2 for all irrational frequencies, solving a long-standing problem. Other corollaries include two very elementary proofs of zero measure of the spectrum (Problem 5 in [41]) and a similar Hausdorff dimension result for the quantum graph graphene.
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Cantor spectrum for multidimensional quasi-periodic Schr\"odinger operators
A dense, positive-Hausdorff-dimension set of frequencies gives the multidimensional almost Mathieu operator a zero-measure Cantor spectrum, and critical almost Mathieu spectra can have arbitrarily small positive box dimension.