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.
Avila, Global theory of one-frequency Schr¨ odinger operators, Acta Math
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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.