Consider a finite polysquare or square tiled region, a connected, but not necessarily simply-connected, polygonal region tiled with aligned unit squares. Using ideas from diophantine approximation, we prove that a half-infinite billiard orbit in such a region is superdense, a best possible form of time-quantitative density, if and only if the initial slope of the orbit is a badly approximable number. As the traditional approach to questions of density and uniformity via ergodic theory depends on results such as Birkhoffs ergodic theorem which are essentially time-qualitative in nature and do not appear to lead naturally to time-quantitative statements, we appeal to a non-ergodic approach that is based largely on number theory and combinatorics. In particular, we use the famous 3-distance theorem in diophantine approximation combined with an iterative process. This paper improves on an earlier result of the authors and Yang where it is shown that badly approximable numbers that satisfy a quite severe technical restriction on the digits of their continued fractions lead to superdensity. Here we overcome this technical impediment.
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