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We disprove a conjecture of Breuer-Last-Simon concerning the absolutelycontinuous spectrum of Jacobi matrices with coefficients that obey an $\ell^2$bounded variation condition with step $q$. We prove existence of a.c. spectrumon a smaller set than that specified by the conjecture and prove that ourresult is optimal.

$\ell^2$ bounded variation and absolutely continuous spectrum of Jacobi matrices