McShane-Rivin norm balls and simple-length multiplicities
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911685500993536 |
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| author | Doan, Nhat Minh Li, Xiaobin Nguyen, Van |
| author_facet | Doan, Nhat Minh Li, Xiaobin Nguyen, Van |
| contents | We use normal-turn estimates for McShane--Rivin norm balls to prove that, for every complete finite-area hyperbolic once-punctured torus $X$, the number of simple closed geodesics of length exactly $L\geq 2$ is at most $C_X(\log L)^2$. For the modular torus, this gives $$ \#λ_M^{-1}(m)\leq C(\log\log(3m))^2 $$ for every Markoff number $m$, improving the previous logarithmic Markoff-fiber bounds. These estimates also give new quantitative information on the local geometry of McShane--Rivin norm balls, including obstructions to infinite-order flatness at certain irrational directions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_14574 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | McShane-Rivin norm balls and simple-length multiplicities Doan, Nhat Minh Li, Xiaobin Nguyen, Van Geometric Topology Metric Geometry Number Theory Primary 30F60, Secondary 57K20, 52A10, 11H06, 11J06 We use normal-turn estimates for McShane--Rivin norm balls to prove that, for every complete finite-area hyperbolic once-punctured torus $X$, the number of simple closed geodesics of length exactly $L\geq 2$ is at most $C_X(\log L)^2$. For the modular torus, this gives $$ \#λ_M^{-1}(m)\leq C(\log\log(3m))^2 $$ for every Markoff number $m$, improving the previous logarithmic Markoff-fiber bounds. These estimates also give new quantitative information on the local geometry of McShane--Rivin norm balls, including obstructions to infinite-order flatness at certain irrational directions. |
| title | McShane-Rivin norm balls and simple-length multiplicities |
| topic | Geometric Topology Metric Geometry Number Theory Primary 30F60, Secondary 57K20, 52A10, 11H06, 11J06 |
| url | https://arxiv.org/abs/2605.14574 |