Gamburd, Alexander; Magee, Michael; Ronan, Ryan An asymptotic formula for integer points on Markoff-Hurwitz varieties. (English) Zbl 1447.11051 Ann. Math. (2) 190, No. 3, 751-809 (2019). 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. Reviewer: Arthur Baragar (Las Vegas) Cited in 8 Documents 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 Keywords:Markoff-Hurwitz equations; counting integer points on varieties; uniformly contracting dynamics; conformal measures Citations:Zbl 0501.10015; Zbl 0892.11009; Zbl 0255.52006; Zbl 1235.22015 × Cite Format Result Cite Review PDF Full Text: DOI arXiv Link References: [1] Avila, Artur; Hubert, Pascal; Skripchenko, Alexandra, Diffusion for chaotic plane sections of 3-periodic surfaces, Invent. Math.. 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