  We apply the methods of classical approximation theory (extreme properties of
polynomials) to study the essential support $\Sigma_{ac}$ of the absolutely
continuous spectrum of Jacobi matrices. First, we prove an upper bound on the
measure of $\Sigma_{ac}$ 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.
