Limit-periodic Dirac operators with thin spectra

We prove that limit-periodic Dirac operators generically have spectra of zero Lebesgue measure and a dense set them Hausdorff dimension. The proof combines ideas Avila from Schrödinger setting with new commutation argument for generating open spectral gaps. This overcomes an obstacle previously observed in the literature; namely, Schrödinger-type settings, translation corresponds to small L?-perturbations operator data, but this is not true or CMV operators. much more model-independent. To demonstrate this, we also apply generic zero-measure spectrum matrices Verblunsky coefficients.