2021/06/26 by József Beck, Beck, J., W.W.L. Chen +3
Mathematics · Physics and Astronomy · #11K38 #37E35 #Advanced Mathematical Theories and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2106.14001
openalex publication_date 2021/06/26 · openalex created_date 2021/11/08 · openalex updated_date 2026/07/28
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 \qα\, q=0,1,2,3,…. If α is badly approximable and b∈(0,1) satisfies b≠\mα\ for any m∈ℤ, then the parity of cardinalities of the sets \1≤ q≤ N:\qα\∈[0,b)\ as N→∞ 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 α and b=\mα\ for some m∈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 α-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).