Stable-norm boundary geometry on once-punctured tori
Abstract
We use normal-turn estimates to study the local and global geometry of the boundaries of stable-norm balls BX for complete finite-area hyperbolic once-punctured tori X. We show that the boundary ∂ BX is a convex-geometric detector of exponential Diophantine approximation: rational directions give genuine corners with exponentially small exterior angles, while at an irrational direction β the graph-flatness order admits an explicit formula in terms of the exponential rate at which rational directions approach β and the ℓ∞-radius of BX in the projective direction β. Thus irrational directions are not uniformly flat to infinite order, correcting the McShane--Rivin local picture. We also determine all possible irrational flatness orders and the size of the corresponding level sets. The computation of the rational corners unexpectedly yields a new McShane-type identity for the area of the dual body BX*: Area(BX*) = 4∑γ ℓX(γ) logcoth(ℓX(γ)/2), where the sum is over unoriented simple closed geodesics on X. As another global application, the normal-turn estimates also give a logarithmic-square bound for the number of integer points on the boundary of each dilated norm ball. Consequently, the number of simple closed geodesics of length exactly L≥2 is at most CX(log L)2. For the modular torus this yields #λM−1(m)≤ C(loglog(3m))2 for every Markoff number m, improving the previous logarithmic bounds for Markoff fibers.
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BibTeX
@misc{doan2026stablenorm,
title = {{Stable-norm boundary geometry on once-punctured tori}},
author = {Nhat Minh Doan and Xiaobin Li and Van Nguyen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.14574},
eprint = {2605.14574},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.14574},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Stable-norm boundary geometry on once-punctured tori
AU - Nhat Minh Doan
AU - Xiaobin Li
AU - Van Nguyen
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2605.14574
UR - https://arxiv.org/abs/2605.14574
AB - We use normal-turn estimates to study the local and global geometry of the boundaries of stable-norm balls B_X for complete finite-area hyperbolic once-punctured tori X. We show that the boundary ∂ B_X is a convex-geometric detector of exponential Diophantine approximation: rational directions give genuine corners with exponentially small exterior angles, while at an irrational direction β the graph-flatness order admits an explicit formula in terms of the exponential rate at which rational directions approach β and the ℓ^∞-radius of B_X in the projective direction β. Thus irrational directions are not uniformly flat to infinite order, correcting the McShane--Rivin local picture. We also determine all possible irrational flatness orders and the size of the corresponding level sets. The computation of the rational corners unexpectedly yields a new McShane-type identity for the area of the dual body B_X^*: Area(B_X^*) = 4∑_γ ℓ_X(γ) logcoth(ℓ_X(γ)/2), where the sum is over unoriented simple closed geodesics on X. As another global application, the normal-turn estimates also give a logarithmic-square bound for the number of integer points on the boundary of each dilated norm ball. Consequently, the number of simple closed geodesics of length exactly L≥2 is at most C_X(log L)^2. For the modular torus this yields #λ_M^−1(m)≤ C(loglog(3m))^2 for every Markoff number m, improving the previous logarithmic bounds for Markoff fibers.
ER -