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 ${\mathrm{\Sigma }}_{ac}$ of the absolutely continuous spectrum of Jacobi matrices. First, we prove an upper bound on the measure of ${\mathrm{\Sigma }}_{ac}$ 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.