## Zero measure spectrum for the almost Mathieu operator.(English)Zbl 0814.11040

The almost Mathieu (or Harper’s) operator on $$\ell^ 2(\mathbb{Z})$$ is given by $(H_{\alpha,\lambda,\theta}u)(n)= u(n + 1) + u(n -1) + \lambda \cos (2\pi\alpha n + \theta) u(n)$ and is related to Schrödinger’s operator and to the rotation number. Upper and lower estimates for the Lebesgue measure of the spectrum of the operator are obtained for each $$\lambda$$, $$\theta$$, $$\alpha$$. The measure is deduced to be $$| 4- | \lambda||$$ for all $$\lambda$$, $$\theta$$ and for $$\alpha$$ with unbounded partial quotients, thus for all $$\lambda$$, $$\theta$$ and for almost all $$\alpha$$. When $$| \lambda| = 2$$, the spectrum is a Cantor set of measure 0. In an interesting mix of operator and number theory, it is further shown that the Hausdorff dimension of the spectrum is at most 1/2 for irrational $$\alpha$$ such that for any $$c > 0$$, there are infinitely many rationals $$p_ n/q_ n$$ satisfying $\left | \alpha - {p_ n\over q_ n}\right | < {c\over q_ n^ 4},$ a set of dimension 1/2 (for $$\alpha$$ with unbounded quotients, the denominator on the RHS is $$q^ 2_ n)$$.

### MSC:

 11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension 34L05 General spectral theory of ordinary differential operators 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 70H99 Hamiltonian and Lagrangian mechanics
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### References:

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