  We introduce a new method to establish time-quantitative density in flat
dynamical systems. First we give a shorter and different proof of our earlier
result that a half-infinite geodesic on an arbitrary finite polysquare surface
P is superdense on P if the slope of the geodesic is a badly approximable
number. We then adapt our method to study time-quantitative density of
half-infinite geodesics on algebraic polyrectangle surfaces.
