New Kronecker-Weyl type equidistribution results and diophantine approximation

Abstract

An interesting result of Veech more than 50 years ago is a parity, or mod $2$, version of the Kronecker--Weyl equidistribution theorem concerning the irrational rotation sequence $\{q\hat{I}\pm \}$, $q=0,1,2,3,\dots .$ If $\hat{I}\pm$ is badly approximable and $b\in (0,1)$ satisfies $b\ne \{m\hat{I}\pm \}$ for any $m\in \mathbb{Z}$, then the parity of cardinalities of the sets $\{1\le q\le N:\{q\hat{I}\pm \}\in [0,b)\}$ as $N\to \infty$ is evenly distributed. We first answer a question of Veech and establish a stronger form of the mod $n$ analog of his result (Theorem 3.1). Furthermore, for irrational $\hat{I}\pm$ and $b=\{m\hat{I}\pm \}$ for some $m\in \mathbb{N}$, we give a simple yet precise characterization of those cases that give rise to even distribution (Theorem 2.1). We also obtain time-quantitative description of some very striking violations of uniformity -- this part is particularly number theoretic in nature, and involves Ostrowski representations of positive integers and $\hat{I}\pm$-expansions of real numbers (Theorem 3.4). The Veech discrete $2$-circle problem can also be visualized as a problem that concerns $1$-direction geodesic flow on a surface obtained by modifying the surface comprising two side-by-side squares by the inclusion of symmetric barriers and gates on the vertical edges, with appropriate modification of the vertical edge identifications. We establish a far-reaching generalization of this case to ones that concern $1$-direction geodesic flow on surfaces obtained by modifying a finite square tiled translation surface in analogous but not necessarily symmetric ways (Theorem 3.2).

70 pages, 27 figures