Generalization of a density theorem of Khinchin and diophantine approximation

TL;DR

This paper extends Khinchin's density theorem from the unit square to finite polysquare surfaces, using number theory tools like Diophantine approximation and continued fractions, to characterize superdense geodesics.

Contribution

It generalizes Khinchin's result to more complex surfaces, removing previous technical restrictions and providing a self-contained number-theoretic proof.

Findings

Superdensity characterized by badly approximable slopes on polysquare surfaces.

Extension of Khinchin's theorem to finite tiled surfaces.

Overcoming previous technical restrictions on continued fraction digits.

Abstract

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.