  Based on a thorough numerical analysis of the spectrum of Harper's operator,
which describes, e.g., an electron on a two-dimensional lattice subjected to a
magnetic field perpendicular to the lattice plane, we make the following
conjecture: For any value of the incommensurability parameter sigma of the
operator its spectrum can be covered by the bands of the spectrum for every
rational approximant of sigma after stretching them by factors with a common
upper bound. We show that this conjecture has the following important
consequences: For all irrational values of sigma the spectrum is (i) a zero
measure Cantor set and has (ii) a Hausdorff dimension less or equal to 1/2. We
propose that our numerical approach may be a guide in finding a rigorous proof
of these results.
