Log-Hausdorff multifractality of the absolutely continuous spectral measure of the almost Mathieu operator

Abstract: This paper focuses on the fractal characteristics of the absolutely continuous spectral measure of the subcritical almost Mathieu operator (AMO) and Diophantine frequency. In particular, we give a complete description of the (classical) multifractal spectrum and a finer description in the logarithmic gauge. The proof combines continued$-$fraction$/$metric Diophantine techniques and refined covering arguments. These results rigorously substantiate (and quantify in a refined gauge) the physicists' intuition that the absolutely continuous component of the spectrum is dominated by energies with trivial scaling index, while also exhibiting nontrivial exceptional sets which are negligible for classical Hausdorff measure but large at the logarithmic scale.