Generalization of a density theorem of Khinchin and diophantine approximation

The continuous version of a famous result of Khinchin says that a half-infinite torus line in the unit square [0,1] 2 exhibits superdensity, a best form of time-quantitative density, if and only if the slope of the geodesic is a badly approximable number. We extend this result of Khinchin to the case when the unit torus [0,1] 2 is replaced by a finite polysquare translation surface, or square tiled surface. In particular, we show that it is possible to study this very number-theoretic problem by restricting to traditional tools in number theory, using only continued fractions and the famous 3-distance theorem in diophantine approximation combined with an iterative process.