{"paper":{"title":"McShane-Rivin norm balls and simple-length multiplicities","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"For any finite-area hyperbolic once-punctured torus, at most C (log L)^2 simple closed geodesics share any fixed length L.","cross_cats":["math.MG","math.NT"],"primary_cat":"math.GT","authors_text":"Nhat Minh Doan, Van Nguyen, Xiaobin Li","submitted_at":"2026-05-14T08:45:18Z","abstract_excerpt":"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 $$ \\#\\lambda_M^{-1}(m)\\leq C(\\log\\log(3m))^2 $$ 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."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For every complete finite-area hyperbolic once-punctured torus X, 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 (log log(3m))^2.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The normal-turn estimates for McShane-Rivin norm balls hold uniformly enough to convert growth control into a multiplicity bound without additional length-dependent losses.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"On hyperbolic once-punctured tori the multiplicity of any simple geodesic length L is at most C (log L)^2, improving prior logarithmic bounds for Markoff numbers.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"For any finite-area hyperbolic once-punctured torus, at most C (log L)^2 simple closed geodesics share any fixed length L.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"b909bb63ca3b6353aa97cd80e419b0069b05a27220094c758332e7e4b24c28ab"},"source":{"id":"2605.14574","kind":"arxiv","version":1},"verdict":{"id":"38f99c47-95c0-4487-baa7-e7756e7c0dd4","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T01:18:59.552058Z","strongest_claim":"For every complete finite-area hyperbolic once-punctured torus X, 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 (log log(3m))^2.","one_line_summary":"On hyperbolic once-punctured tori the multiplicity of any simple geodesic length L is at most C (log L)^2, improving prior logarithmic bounds for Markoff numbers.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The normal-turn estimates for McShane-Rivin norm balls hold uniformly enough to convert growth control into a multiplicity bound without additional length-dependent losses.","pith_extraction_headline":"For any finite-area hyperbolic once-punctured torus, at most C (log L)^2 simple closed geodesics share any fixed length L."},"references":{"count":36,"sample":[{"doi":"","year":2013,"title":"2013 , doi =","work_id":"bd2fe240-a42e-4386-b357-187b71137dd1","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Khinchin, Aleksandr Yakovlevich , title =","work_id":"f4ba514a-1ff0-45c7-8a95-9660fdbff278","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Rockett, Andrew M. and Sz. Continued Fractions , publisher =","work_id":"36d8cfe7-700d-4b58-a5ca-0ef8ea8bc791","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Durrett, Rick , title =","work_id":"ed931e10-7526-4a75-b69d-80500362fc1b","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Cassels, J. W. S. , title =","work_id":"6b26e16e-5aa1-4b10-bca1-b4a2c888c8d2","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":36,"snapshot_sha256":"e4f8c93d56c847cff635043e5a9c169ca5373d492de84b60a1a7a5b669b620c5","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}