Generalization of a density theorem of Khinchin and diophantine approximation

The continuous version of a fundamental result of Khinchin says that a half-infinite torus line in the unit square $[0,1]^2$ exhibits superdensity, which is a best form of time-quantitative density, if and only if the slope of the geodesic is a badly approximable number. In this paper, we give a proof of the extension of this result of Khinchin to the case when the unit torus $[0,1]^2$ is replaced by a finite polysquare surface, or square tiled surface. The argument is based on diophantine approximation and continued fractions, traditional tools in number theory. In particular, we use the famous $3$-distance theorem in diophantine approximation combined with an iterative process. In short, this is a very number-theoretic study of a very number-theoretic problem. 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 superdense geodesics. Here we overcome this technical impediment. This paper is self-contained, and the reader does not need any knowledge of dynamical systems.