On the Measure of the Absolutely Continuous Spectrum for Jacobi Matrices

We apply the methods of classical approximation theory (extreme properties of polynomials) to study the essential support $Σ_{ac}$ of the absolutely continuous spectrum of Jacobi matrices. First, we prove an upper bound on the measure of $Σ_{ac}$ which takes into account the value distribution of the diagonal elements, and implies the bound due to Deift-Simon and Poltoratski-Remling. Second, we generalise the differential inequality of Deift-Simon for the integrated density of states associated with the absolutely continuous spectrum to general Jacobi matrices.

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