ZERO MEASURE SPECTRUM FOR THE ALMOST MATHIEU OPERATOR

We study the almost Mathieu operator: (H(alpha, lambda, theta(u))(n) = u(n + 1) + u(n - 1) + lambda cos(2pialphan + theta)u(n), on l2(Z), and show that for all lambda, theta, and (Lebesgue) a.e. alpha, the Lebesgue measure of its spectrum is precisely \4 - 2\lambda\\. In particular, for \lambda\ = 2 the spectrum is a zero measure cantor set. Moreover, for a large set of irrational alpha's (and \lambda\ = 2) we show that the Hausdorff dimension of the spectrum is smaller than or equal to 1/2.