No abstract

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

In 1935, Besicovitch proved a remarkable theorem indicating that an integrable function f on R 2 is strongly differentiable if and only if its associated strong maximal function M S f is finite a.e. We consider analogues of Besicovitch's result in the context of ergodic theory, in particular discussing the problem of whether or not, given a (not necessarily integrable) measurable function f on a nonatomic probability space and a measure preserving transformation T on that space, the ergodic averages of f with respect to T converge a.e. if and only if the associated ergodic maximal function T * f is finite a.e. Of particular relevance to this discussion will be recent results in the field of inhomogeneous diophantine approximation. Let f be an integrable function on R 2. A classical result in analysis, the Lebesgue Differentiation Theorem, tells us that, for a.e. x ∈ R 2 , the averages of f over disks shrinking to x tend to f (x) itself. More precisely, we have that lim r→0