  It has been conjectured that for a class of piecewise linear maps the closure
of the set of images of the discontinuity has the structure of a fat fractal,
that is, a fractal with positive measure. An example of such maps is the
sawtooth map in the elliptic regime. In this work we analyze this problem
quantum mechanically in the semiclassical regime. We find that the fraction of
states localized on the unstable set satisfies a modified fractal Weyl law,
where the exponent is given by the exterior dimension of the fat fractal.
