Cantor spectrum for multidimensional quasi-periodic Schrödinger operators

In this paper, we prove that for a dense set of irrational frequencies with positive Hausdorff dimension, the Hausdorff (and upper box) dimension of the spectrum of the critical almost Mathieu operator is positive, yet can be made arbitrarily small. As a consequence, we investigate the spectrum of a class of multidimensional quasi-periodic Schrödinger operators that exhibit a Cantor spectrum, which answers a question posed by Damanik, Fillman, and Gorodetski [24].