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In the spectral theory of orthogonal polynomials (or Jacobi matrices) the goal is to relate information about the recurrence coefficients (=Jacobi parameters) to information about the measure of orthogonality (=spectral measure). When the essential spectrum is an interval, the situation is well understood. But what happens when gaps occur? In the talk I'll discuss results for Jacobi matrices whose essential spectrum is a finite union of closed intervals. In particular, I'll show how to obtain Szegö's theorem from a non-local step-by-step sum rule. Moreover, I'll explain how one can use results of Remling and Sodin-Yuditskii to prove that the Jacobi parameters approach a single point on the isospectral torus in the setting of Szegö's theorem. The talk is based on joint work with Barry Simon and Maxim Zinchenko. Our results rely on potential theory and analytic function theory, with an important link to Riemann surfaces and Fuchsian groups.