An asymptotic formula for integer points on Markoff-Hurwitz varieties

Alex Gamburd; Michael Magee; Ryan Ronan

doi:https://doi.org/10.4007/annals.2019.190.3.2

Abstract

We establish an asymptotic formula for the number of integer solutions to the Markoff-Hurwitz equation \[ x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}=ax_{1}x_{2}\cdots x_{n}+k. \] When $n\geq 4$, 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.

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@BOOK{MCSHANETHESIS, author = {McShane, Greg}, title = {A Remarkable Identity for Lengths of Curves}, note = {Ph.D. thesis, University of Warwick, UK}, publisher = {ProQuest LLC, Ann Arbor, MI}, year = {1991}, pages = {1}, mrclass = {Thesis}, mrnumber = {3389436}, url = {http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:U637636}, zblnumber = {}, }
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@ARTICLE{MCSHANEIDENTITY, author = {McShane, Greg}, title = {Simple geodesics and a series constant over {T}eichmuller space}, journal = {Invent. Math.}, fjournal = {Inventiones Mathematicae}, volume = {132}, year = {1998}, number = {3}, pages = {607--632}, issn = {0020-9910}, mrclass = {32G15 (30F60 57M50)}, mrnumber = {1625712}, mrreviewer = {Thomas A. Schmidt}, doi = {10.1007/s002220050235}, url = {https://doi.org/10.1007/s002220050235}, zblnumber = {0916.30039}, }
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@ARTICLE{MIRZSIMPLE, author = {Mirzakhani, Maryam}, title = {Growth of the number of simple closed geodesics on hyperbolic surfaces}, journal = {Ann. of Math. (2)}, fjournal = {Annals of Mathematics. Second Series}, volume = {168}, year = {2008}, number = {1}, pages = {97--125}, issn = {0003-486X}, mrclass = {32G15}, mrnumber = {2415399}, mrreviewer = {Hsian-Hua Tseng}, doi = {10.4007/annals.2008.168.97}, url = {https://doi.org/10.4007/annals.2008.168.97}, zblnumber = {1177.37036}, }
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@ARTICLE{MORDELL, author = {Mordell, L. J.}, title = {On the integer solutions of the equation {$x^2+y^2+z\sp 2+2xyz=n$}}, journal = {J. London Math. Soc.}, fjournal = {Journal of the London Mathematical Society. Second Series}, volume = {28}, year = {1953}, pages = {500--510}, issn = {0024-6107}, mrclass = {10.0X}, mrnumber = {0056619}, mrreviewer = {I. Niven}, doi = {10.1112/jlms/s1-28.4.500}, url = {https://doi.org/10.1112/jlms/s1-28.4.500}, zblnumber = {0051.27802}, }
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@ARTICLE{RIVINMCSHANE, author = {McShane, Greg and Rivin, Igor}, title = {A norm on homology of surfaces and counting simple geodesics}, journal = {Internat. Math. Res. Notices}, fjournal = {International Mathematics Research Notices}, year = {1995}, number = {2}, pages = {61--69}, issn = {1073-7928}, mrclass = {57M50 (11J06 57N05)}, mrnumber = {1317643}, mrreviewer = {Lee Mosher}, doi = {10.1155/S1073792895000055}, url = {https://doi.org/10.1155/S1073792895000055}, zblnumber = {0828.30023}, }
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@ARTICLE{NOV, author = {Novikov, S. P.}, title = {The {H}amiltonian formalism and a multivalued analogue of {M}orse theory}, journal = {Uspekhi Mat. Nauk}, fjournal = {Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk}, volume = {37}, year = {1982}, number = {5(227)}, pages = {3--49, 248}, issn = {0042-1316}, mrclass = {58E05 (58E30 58F05)}, mrnumber = {0676612}, mrreviewer = {P. Ver Eecke}, zblnumber = {0571.58011}, doi = {10.1070/RM1982v037n05ABEH004020}, url = {https://doi.org/10.1070/RM1982v037n05ABEH004020}, }
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@ARTICLE{PATTERSON, author = {Patterson, S. J.}, title = {The limit set of a {F}uchsian group}, journal = {Acta Math.}, fjournal = {Acta Mathematica}, volume = {136}, year = {1976}, number = {3-4}, pages = {241--273}, issn = {0001-5962}, mrclass = {30A58 (10D10 20H10)}, mrnumber = {0450547}, mrreviewer = {B. Maskit}, doi = {10.1007/BF02392046}, url = {https://doi.org/10.1007/BF02392046}, zblnumber = {0336.30005}, }
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@MISC{POLLICOTTRVZ, author = {Pollicott, Mark}, title = {Apollonian circle packings}, note = {available at the author's homepage}, year = {2014}, zblnumber = {1338.52022}, }
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@ARTICLE{PP2, author = {Parry, William and Pollicott, Mark}, title = {An analogue of the prime number theorem for closed orbits of {A}xiom {A} flows}, journal = {Ann. of Math. (2)}, fjournal = {Annals of Mathematics. Second Series}, volume = {118}, year = {1983}, number = {3}, pages = {573--591}, issn = {0003-486X}, mrclass = {58F22 (58F20)}, mrnumber = {0727704}, mrreviewer = {Peter Sarnak}, doi = {10.2307/2006982}, url = {https://doi.org/10.2307/2006982}, zblnumber = {0537.58038}, }
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@book{PP, author = {Parry, William and Pollicott, Mark}, title = {Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics}, series = {Astérisque}, fjournal = {Astérisque}, volume = {187-188}, year = {1990}, publisher={Soc. Math. France, Paris}, pages = {268}, issn = {0303-1179}, mrclass = {58F20 (58F11 58F15)}, mrnumber = {1085356}, mrreviewer = {Nicolaĭ T. A. Haydn}, zblnumber = {0726.58003}, }
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@INCOLLECTION{Silverman, author = {Silverman, Joseph H.}, title = {Counting Integer and Rational Points on Varieties}, booktitle = {Columbia University Number Theory Seminar}, venue = {New York, 1992}, series = {Astérisque}, publisher = {Math. Soc. France, Paris}, volume = {228}, year = {1995}, pages = {4, 223--236}, issn = {0303-1179}, mrclass = {11G35 (11D72 14G05)}, mrnumber = {1330936}, mrreviewer = {Takeshi Ooe}, zblnumber = {0834.11029}, url = {http://www.numdam.org/item/AST_1995__228__223_0/}, }
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@ARTICLE{SM, author = {Schwartz, H. and Muhly, H. T.}, title = {On a class of cubic {D}iophantine equations}, journal = {J. London Math. Soc.}, fjournal = {Journal of the London Mathematical Society. Second Series}, volume = {32}, year = {1957}, pages = {379--382}, issn = {0024-6107}, mrclass = {10.0X}, mrnumber = {0091292}, mrreviewer = {H. W. Brinkmann}, doi = {10.1112/jlms/s1-32.3.379}, url = {https://doi.org/10.1112/jlms/s1-32.3.379}, zblnumber = {0080.26001}, }
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@ARTICLE{SUL79, author = {Sullivan, Dennis}, title = {The density at infinity of a discrete group of hyperbolic motions}, journal = {Inst. Hautes \'{E}tudes Sci. Publ. Math.}, fjournal = {Institut des Hautes \'{E}tudes Scientifiques. Publications Mathématiques}, number = {50}, year = {1979}, pages = {171--202}, issn = {0073-8301}, mrclass = {58F17 (22E40 28C10 30C85)}, mrnumber = {0556586}, mrreviewer = {Troels J\o rgensen}, url = {http://www.numdam.org/item?id=PMIHES_1979__50__171_0}, zblnumber = {0439.30034}, }
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@ARTICLE{ZAGIER, author = {Zagier, Don}, title = {On the number of {M}arkoff numbers below a given bound}, journal = {Math. Comp.}, fjournal = {Mathematics of Computation}, volume = {39}, year = {1982}, number = {160}, pages = {709--723}, issn = {0025-5718}, mrclass = {10F20 (10A20 10B10)}, mrnumber = {0669663}, mrreviewer = {Richard T. Bumby}, doi = {10.2307/2007348}, url = {https://doi.org/10.2307/2007348}, zblnumber = {0501.10015}, }
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@INCOLLECTION{zorich, author = {Zorich, Anton}, title = {Flat surfaces}, booktitle = {Frontiers in Number Theory, Physics, and Geometry. {I}}, pages = {437--583}, publisher = {Springer, Berlin}, year = {2006}, mrclass = {37D40 (30F30 32G15 37D50 57M50)}, mrnumber = {2261104}, mrreviewer = {Thomas A. Schmidt}, arxiv = {math/0609392}, zblnumber = {1129.32012}, }

An asymptotic formula for integer points on Markoff-Hurwitz varieties

Abstract

Authors