{"paper":{"title":"An asymptotic formula for integer points on Markoff-Hurwitz varieties","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.GR"],"primary_cat":"math.NT","authors_text":"Alex Gamburd, Michael Magee, Ryan Ronan","submitted_at":"2016-03-20T20:46:53Z","abstract_excerpt":"We establish an asymptotic formula for the number of integer solutions to the Markoff-Hurwitz equation \\[ x_{1}^{2}+x_{2}^{2}+\\ldots+x_{n}^{2}=ax_{1}x_{2}\\ldots x_{n}+k. \\] When $n\\geq4$ the previous best result is by Baragar (1998) that gives an exponential rate of growth with exponent $\\beta$ that is not in general an integer when $n\\geq 4$. We give a new interpretation of this exponent of growth in terms of the unique parameter for which there exists a certain conformal measure on projective space."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.06267","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","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"}