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An asymptotic formula for integer points on Markoff-Hurwitz varieties. (English) Zbl 1447.11051

In this paper, the authors consider the variety \(V\) defined by the generalized Markoff-Hurwitz equation \(x_1^2+ x_2^2+\cdots +x_n^2=ax_1x_2\cdots x_n+k\) and seek the rate of growth for the set \(V(\mathbb Z)\cap B(R)\), where \(B(R)\) is a ball of radius \(R\). For the Markoff equation (\(n=3\), \(a=3\), \(k=0\)), D. Zagier [Math. Comput. 39, 709–723 (1982; Zbl 0501.10015)] showed \(|V(\mathbb Z)\cap B(R)|=c(\log R)^2+O(\log R(\log\log R)^2)\) and calculated \(c\). The case \(n\geq 4\) (with \(k=0\)) was studied by the reviewer, who showed [Pac. J. Math. 182, No. 1, 1–21 (1998; Zbl 0892.11009)] there exists a constant \(\beta=\beta(n)\) so that if \(V(\mathbb Z)\) includes a non-trivial solution (i.e. not all zeros), then \(|V(\mathbb Z)\cap B(R)|=(\log R)^{\beta+o(1)}\), and found bounds on \(\beta(n)\). Except for \(\beta(3)=2\) (by Zagier’s result), the exact value of \(\beta(n)\) is not known. The authors improve Baragar’s result to a true asymptotic: For each \((n,a,k)\) with \(V(\mathbb Z)-{\mathcal E}\) infinite, there is a positive \(c=c(n,a,k)\) such that \[ |V(\mathbb Z)\cap B(R)|=c(\log R)^\beta + o((\log R)^\beta), \] where \(\beta=\beta(n)\) depends only on \(n\). The set \({\mathcal E}\) is a certain exceptional family of solutions. It is non-empty only if \(k-n+2\) or \(k-n-1\) is a square, and in those cases, it dominates, as \({\mathcal E}\cap B(R)|\geq cR\) for some \(c>0\). (The exceptional set \({\mathcal E}\) lies on an infinite union of rational curves on the variety, one of which is a line.)
There is a nice parallel with a similar problem related to Apollonian circle packings: Let \(N(R)\) count the number circles with curvature bounded by \(R\) in some Apollonian circle packing. D. W. Boyd [Aequationes Math. 9, 99–106 (1973; Zbl 0255.52006)] found \(N(R)=R^{\delta+o(1)}\) where \(\delta\) is independent of the packing (it is the Hausdorff dimension of the residual set). A. Kontorovich and H. Oh [J. Am. Math. Soc. 24, No. 3, 603–648 (2011; Zbl 1235.22015)] improve this to a true asymptotic.

MSC:

11D45 Counting solutions of Diophantine equations
11D25 Cubic and quartic Diophantine equations
11D41 Higher degree equations; Fermat’s equation
11J70 Continued fractions and generalizations
37D25 Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
37D35 Thermodynamic formalism, variational principles, equilibrium states for dynamical systems

References:

[1] Avila, Artur; Hubert, Pascal; Skripchenko, Alexandra, Diffusion for chaotic plane sections of 3-periodic surfaces, Invent. Math.. Inventiones Mathematicae, 206, 109-146 (2016) · Zbl 1376.37030 · doi:10.1007/s00222-016-0650-z
[2] Avila, Artur; Hubert, Pascal; Skripchenko, Alexandra, On the {H}ausdorff dimension of the {R}auzy gasket, Bull. Soc. Math. France. Bulletin de la Soci\'{e}t\'{e} Math\'{e}matique de France, 144, 539-568 (2016) · Zbl 1356.37018 · doi:10.24033/bsmf.2722
[3] Aigner, Martin, Markov’s Theorem and 100 Years of the Uniqueness Conjecture: A Mathematical Journey from Irrational Numbers to Perfect Matchings, x+257 pp. (2013) · Zbl 1276.00006 · doi:10.1007/978-3-319-00888-2
[4] Arnoux, Pierre; Starosta, {\v{S}}t{\v{e}}p{\'{a}}n, The {R}auzy gasket. Further Developments in Fractals and Related Fields, Trends Math., 1-23 (2013) · Zbl 1268.28007 · doi:10.1007/978-0-8176-8400-6_1
[5] Baladi, Viviane, Positive Transfer Operators and Decay of Correlations, Adv. Ser. Nonlinear Dynam., 16, x+314 pp. (2000) · Zbl 1012.37015 · doi:10.1142/9789812813633
[6] Baragar, Arthur, Asymptotic growth of {M}arkoff-{H}urwitz numbers, Compositio Math.. Compositio Mathematica, 94, 1-18 (1994) · Zbl 0813.11014
[7] Baragar, Arthur, Integral solutions of {M}arkoff-{H}urwitz equations, J. Number Theory. Journal of Number Theory, 49, 27-44 (1994) · Zbl 0820.11016 · doi:10.1006/jnth.1994.1078
[8] Baragar, Arthur, The exponent for the {M}arkoff-{H}urwitz equations, Pacific J. Math.. Pacific Journal of Mathematics, 182, 1-21 (1998) · Zbl 0892.11009 · doi:10.2140/pjm.1998.182.1
[9] Belyi, G. V., Markov’s Numbers and Quadratic Forms, J. Math. Sci. (New York). Journal of Mathematical Sciences (New York), 106, 3087-3097 (2001) · Zbl 1168.11315 · doi:10.1023/A:1011316920093
[10] Bombieri, Enrico, Continued fractions and the {M}arkoff tree, Expo. Math.. Expositiones Mathematicae, 25, 187-213 (2007) · Zbl 1153.11030 · doi:10.1016/j.exmath.2006.10.002
[11] Boyd, David W., The disk-packing constant, Aequationes Math.. Aequationes Mathematicae, 7, 182-193 (1971) · Zbl 0229.52012 · doi:10.1007/BF01818512
[12] Boyd, David W., Improved bounds for the disk-packing constant, Aequationes Math.. Aequationes Mathematicae, 9, 99-106 (1973) · Zbl 0255.52006 · doi:10.1007/BF01838194
[13] Boyd, David W., The sequence of radii of the {A}pollonian packing, Math. Comp.. Mathematics of Computation, 39, 249-254 (1982) · Zbl 0492.52009 · doi:10.2307/2007636
[14] Cassels, J. W. S., An Introduction to {D}iophantine Approximation, Cambridge Tracts in Math. and Math. Phys., 45, x+166 pp. (1957) · Zbl 0077.04801
[15] Cayley, Arthur, A memoir on cubic surfaces, Philosophical Transactions of the Royal Society of London. Philos. Trans. R. Soc. Lond., 159, 231-326 (1869) · JFM 02.0576.01 · doi:10.1098/rstl.1869.0010
[16] De Leo, Roberto, On a generalized {S}ierpinski fractal in {\( \mathbb{R}\text{P}^n\)} (2008)
[17] De Leo, Roberto, A conjecture on the {H}ausdorff dimension of attractors of real self-projective iterated function systems, Exp. Math.. Experimental Mathematics, 24, 270-288 (2015) · Zbl 1404.28009 · doi:10.1080/10586458.2014.987884
[18] DeLeo, Roberto; Dynnikov, Ivan A., Geometry of plane sections of the infinite regular skew polyhedron {\(\{4,6\mid 4\}\)}, Geom. Dedicata. Geometriae Dedicata, 138, 51-67 (2009) · Zbl 1165.28006 · doi:10.1007/s10711-008-9298-1
[19] Gauss, Carl Friedrich, Brief an {L}aplace vom 30 {J}an. 1812, {W}erke \(X_1, 371-374 (1812)\)
[20] Gamburd, A.; Magee, M.; Ronan, R., An asymptotic formula for integer points on {M}arkoff-{H}urwitz varieties (2018)
[21] Goldman, William M., The modular group action on real {\({\rm SL}(2)\)}-characters of a one-holed torus, Geom. Topol.. Geometry and Topology, 7, 443-486 (2003) · Zbl 1037.57001 · doi:10.2140/gt.2003.7.443
[22] Ghosh, A.; Sarnak, P., Integral points on {M}arkoff type cubic surfaces (2017)
[23] Gurwood, Christopher, Diophantine Approximation and the {M}arkov Chain, 68 pp. (1976)
[24] Huang, Yi; Norbury, Paul, Simple geodesics and {M}arkoff quads, Geom. Dedicata. Geometriae Dedicata, 186, 113-148 (2017) · Zbl 1360.30040 · doi:10.1007/s10711-016-0182-0
[25] Horowitz, Robert D., Induced automorphisms on {F}ricke characters of free groups, Trans. Amer. Math. Soc.. Transactions of the American Mathematical Society, 208, 41-50 (1975) · Zbl 0306.20027 · doi:10.2307/1997274
[26] Hu, Hengnan; Tan, Ser Peow; Zhang, Ying, Polynomial automorphisms of {\( \Bbb C^n\)} preserving the {M}arkoff-{H}urwitz polynomial, Geom. Dedicata. Geometriae Dedicata, 192, 207-243 (2018) · Zbl 1387.32027 · doi:10.1007/s10711-017-0235-z
[27] Hurwitz, A., {\"U}ber eine {A}ufgabe der unbestimmten {A}nalysis, Archiv. Math. Phys., 3, 185-196 (1907) · JFM 38.0246.01 · doi:10.1007/978-3-0348-4160-3_27
[28] Ionescu Tulcea, C. T.; Marinescu, G., Th\'{e}orie ergodique pour des classes d’op\'{e}rations non compl\`etement continues, Ann. of Math. (2). Annals of Mathematics. Second Series, 52, 140-147 (1950) · Zbl 0040.06502 · doi:10.2307/1969514
[29] Kato, Tosio, Perturbation Theory for Linear Operators, Grundlehren Math. Wissen., 132, xxi+619 pp. (1976) · Zbl 0836.47009 · doi:10.1007/978-3-642-66282-9
[30] Kontorovich, Alex; Oh, Hee, Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds, J. Amer. Math. Soc.. Journal of the American Mathematical Society, 24, 603-648 (2011) · Zbl 1235.22015 · doi:10.1090/S0894-0347-2011-00691-7
[31] Kuzmin, R. O., On a problem of {G}auss, Atti del Congresso Internazionale dei Matematici, Bologna, 6, 83-89 (1932) · JFM 58.0204.01
[32] Lalley, Steven P., Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-{E}uclidean tessellations and their fractal limits, Acta Math.. Acta Mathematica. Caroline Series, 163, 1-55 (1989) · Zbl 0701.58021 · doi:10.1007/BF02392732
[33] Levitt, Gilbert, La dynamique des pseudogroupes de rotations, Invent. Math.. Inventiones Mathematicae, 113, 633-670 (1993) · Zbl 0791.58055 · doi:10.1007/BF01244321
[34] Liverani, Carlangelo, Decay of correlations, Ann. of Math. (2). Annals of Mathematics. Second Series, 142, 239-301 (1995) · Zbl 0871.58059 · doi:10.2307/2118636
[35] Magee, M., {C}ounting one-sided simple closed geodesics on {F}uchsian thrice punctured projective planes, Internat. Math. Res. Not., 2018, 1-16 (2018) · Zbl 1472.30019 · doi:10.1093/imrn/rny112
[36] Markoff, A., Sur les formes binaires ind\'{e}finies, Math. Ann.. Mathematische Annalen, 15, 381-406 (1879) · JFM 11.0147.01 · doi:10.1007/BF02086269
[37] Markoff, A., Sur les formes quadratiques binaires ind\'{e}finies, Math. Ann.. Mathematische Annalen, 17, 379-399 (1880) · JFM 12.0143.02 · doi:10.1007/BF01446234
[38] McShane, Greg, A Remarkable Identity for Lengths of Curves, 1 pp. (1991)
[39] McShane, Greg, Simple geodesics and a series constant over {T}eichmuller space, Invent. Math.. Inventiones Mathematicae, 132, 607-632 (1998) · Zbl 0916.30039 · doi:10.1007/s002220050235
[40] Mirzakhani, Maryam, Growth of the number of simple closed geodesics on hyperbolic surfaces, Ann. of Math. (2). Annals of Mathematics. Second Series, 168, 97-125 (2008) · Zbl 1177.37036 · doi:10.4007/annals.2008.168.97
[41] Mirzakhani, Maryam, Counting Mapping Class group orbits on hyperbolic surfaces (2016)
[42] Mordell, L. J., On the integer solutions of the equation {\(x^2+y^2+z\sp 2+2xyz=n\)}, J. London Math. Soc.. Journal of the London Mathematical Society. Second Series, 28, 500-510 (1953) · Zbl 0051.27802 · doi:10.1112/jlms/s1-28.4.500
[43] McShane, Greg; Rivin, Igor, A norm on homology of surfaces and counting simple geodesics, Internat. Math. Res. Notices. International Mathematics Research Notices, 61-69 (1995) · Zbl 0828.30023 · doi:10.1155/S1073792895000055
[44] Novikov, S. P., The {H}amiltonian formalism and a multivalued analogue of {M}orse theory, Uspekhi Mat. Nauk. Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk, 37, 3-49 (1982) · Zbl 0571.58011 · doi:10.1070/RM1982v037n05ABEH004020
[45] Patterson, S. J., The limit set of a {F}uchsian group, Acta Math.. Acta Mathematica, 136, 241-273 (1976) · Zbl 0336.30005 · doi:10.1007/BF02392046
[46] 2014, Apollonian circle packings
[47] 2022, Statistical properties of the {R}auzy-{V}eech-{Z}orich map
[48] Pollicott, Mark, A complex {R}uelle-{P}erron-{F}robenius theorem and two counterexamples, Ergodic Theory Dynam. Systems. Ergodic Theory and Dynamical Systems, 4, 135-146 (1984) · Zbl 0575.47009 · doi:10.1017/S0143385700002327
[49] Parry, William; Pollicott, Mark, An analogue of the prime number theorem for closed orbits of {A}xiom {A} flows, Ann. of Math. (2). Annals of Mathematics. Second Series, 118, 573-591 (1983) · Zbl 0537.58038 · doi:10.2307/2006982
[50] Parry, William; Pollicott, Mark, Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics, Ast\'{e}risque. Ast\'{e}risque, 187-188, 268 pp. (1990) · Zbl 0726.58003
[51] Pollicott, M. Urbanski, M., Asymptotic counting in conformal dynamical systems (2017)
[52] Silverman, Joseph H., Integral points on curves and surfaces. Number Theory, Lecture Notes in Math., 1380, 202-241 (1989) · Zbl 0723.14013 · doi:10.1007/BFb0086555
[53] Silverman, Joseph H., Counting Integer and Rational Points on Varieties. Columbia University Number Theory Seminar, Ast\'{e}risque, 228, 4-223 (1995) · Zbl 0834.11029
[54] Schwartz, H.; Muhly, H. T., On a class of cubic {D}iophantine equations, J. London Math. Soc.. Journal of the London Mathematical Society. Second Series, 32, 379-382 (1957) · Zbl 0080.26001 · doi:10.1112/jlms/s1-32.3.379
[55] Sullivan, Dennis, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes \'{E}tudes Sci. Publ. Math.. Institut des Hautes \'{E}tudes Scientifiques. Publications Math\'{e}matiques, 171-202 (1979) · Zbl 0439.30034
[56] Sullivan, Dennis, Entropy, {H}ausdorff measures old and new, and limit sets of geometrically finite {K}leinian groups, Acta Math.. Acta Mathematica. Caroline Series, 153, 259-277 (1984) · Zbl 0566.58022 · doi:10.1007/BF02392379
[57] Wielandt, Helmut, Unzerlegbare, nicht negative {M}atrizen, Math. Z.. Mathematische Zeitschrift, 52, 642-648 (1950) · Zbl 0035.29101 · doi:10.1007/BF02230720
[58] Wirsing, Eduard, On the theorem of {G}auss-{K}usmin-{L}\'{e}vy and a {F}robenius-type theorem for function spaces, Acta Arith.. Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica, 24, 507-528 (1973/74) · Zbl 0283.10032 · doi:10.4064/aa-24-5-507-528
[59] Zagier, Don, On the number of {M}arkoff numbers below a given bound, Math. Comp.. Mathematics of Computation, 39, 709-723 (1982) · Zbl 0501.10015 · doi:10.2307/2007348
[60] Zorich, Anton, Flat surfaces. Frontiers in Number Theory, Physics, and Geometry. {I}, 437-583 (2006) · Zbl 1129.32012
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