Shrinking rates of horizontal gaps for generic translation surfaces

Abstract

A translation surface is given by polygons in the plane, with sides identified by translations to create a closed Riemann surface with a flat structure away from finitely many singular points. Understanding geodesic flow on a surface involves understanding saddle connections. Saddle connections are the geodesics starting and ending at these singular points and are associated to a discrete subset of the plane. To measure the behavior of saddle connections of length at most R, we obtain precise decay rates as R → ∞ for the difference in angle between two almost horizontal saddle connections.Remark 1.2. Note that the choice of a horizontal gap is a convenience. Apply a rotation to the full measure set in Theorem 1.1, and we obtain the same result in a different direction. Consider a countable subset D n = {θ n } n∈N ⊆ [0, 2π). Since a countable union of measure 0 subsets is still measure zero, we obtain a natural corollary that the smallest gap of any of the directions in D n has the same decay rate as given in Theorem 1.1.We first provide an explanation for the scaling factor of R 2 . Masur ([Mas88, Mas90]) showed |Λ ω (R)| has quadratic growth in the sense that for each ω there exist constants c 1 , c 2 so thatfor all large enough R. There are at most 4g − 4 saddle connections in the same direction, so |Θ ω (R)| also has quadratic growth. This result explains the scaling factor of R 2 in Theorem 1.1. The quadratic growth of saddle connections was subsequently built on in [Vee98, EM01, Vor05, EMM15, NRW20].For almost every translation surface, Theorem 1.1 gives partial information on how Θ ω (R) is distributed in [−π, π). For every translation surface, [Mas86] ), then by density the adjacent differences θ j+1 − θ j → 0 as R → ∞ for every ω. Thus ζ ω (R) → 0 as R → ∞ for every ω. However, we cannot expect Theorem 1.1 to hold for every translation surface. Indeed when ω is a lattice surface (see [Mas22, Sections 5 and 7] for a definition), [AC12] showed for every unbounded ψ we have lim infThe results of Theorem 1.1 considers the behavior of a single gap. There is also substantial work done on studying the behavior of the family of gaps. Namely one can study the entire set Θ ω (R). The distribution of normalized gaps exists for almost every ω by [AC12]. In many cases, the distribution has also been computed [ACL15, BMMM + 21, KSW21, UW16, San21]. Considering the behavior of a single gap, which is the focus of the current paper, is orthogonal because the behavior of a single gap does not affect the distribution of gaps.1.1. Outline of proof. The proof follows the now standard strategy of relating a problem about the geometry of a translation surface ω to the orbit of ω under Teichmüller geodesic flow, g t = e t 0 0 e −t . In Section 2 we reduce our problem about gaps to a shrinking target problem for g t . The shrinking targets are sets A t obtained by relating Theorem 1.1 to whether or not g t ω ∈ A t for arbitrarily large t. To prove Theorem 1.1 (2) we use independence results for the sets g −t A t . A key tool to ...