An effective slope gap distribution for lattice surfaces

Tariq Osman; Josh Southerland; Jane Wang

doi:10.3934/dcds.2025081

Article Contents

2025, Volume 45, Issue 12: 4998-5035. Doi: 10.3934/dcds.2025081

This issue Previous Article Indecomposable continua for unbounded itineraries of exponential maps Next Article On the geometry of period doubling invariant sets for area-preserving maps

An effective slope gap distribution for lattice surfaces

1.
Brandeis University, U.S.A
2.
Indiana University, U.S.A
3.
University of Maine, U.S.A

^*Corresponding author: Josh Southerland
^*Corresponding author: Josh Southerland

Received: November 2024

Revised: May 2025

Early access: June 2025

Published: December 2025

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Abstract

We prove an effective slope gap distribution result for translation surfaces whose Veech group is a lattice. As a corollary, we obtain a dynamical proof for an effective gap distribution result for the Farey fractions. As an intermediate step, we prove an effective equidistribution result for the intersection points of long horocycles with a particular transversal of the horocycle flow in $ {\rm SL}_2 (\mathbb{R})/\Gamma $ where $ \Gamma $ is a lattice.

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Mathematics Subject Classification: 37A17, 37D40, 32G15.

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Figure 1. The triangular transversal $ \Omega $ in $ (a,b) $-coordinates, along with the shaded region whose measure gives $ \int_c^d f_{ {\rm gap}}(x) \, dx $

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Figure 2. A parametrization of $ \Omega_i $, where $ c = \alpha_i $ or $ 2\alpha_i $ depending on whether $ -I \in {\rm SL}(X,\omega) $ and the eigenvalues of $ P_i $

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Figure 3. The saddle connections $ \Lambda_R $ on the surface $ (X,\omega) $ on the left and the transformed saddle connections on $ g_{2\log(R)}^{-1}(X,\omega) $ on the right

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