Spectral Homogeneity of Limit-Periodic Schr\"odinger Operators

We prove that the spectrum of a limit-periodic Schr"odinger operator is homogeneous in the sense of Carleson whenever the potential obeys the Pastur–Tkachenko condition. This implies that a dense set of limit-periodic Schr"odinger operators have purely absolutely continuous spectrum supported on a homogeneous Cantor set. When combined with work of Gesztesy–Yuditskii, this also implies that the spectrum of a Pastur–Tkachenko potential has infinite gap length whenever the potential fails to be uniformly almost periodic.