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Limits of translates of divergent geodesics and integral points on one-sheeted hyperboloids. (English) Zbl 1298.22009

Let \(Q\in{\mathbb Z}\left[x_1,x_2,x_3\right]\) be a quadratic form of signature \((2,1)\), and let \(V_m({\mathbb Z})=\left\{x\in{\mathbb Z}^3: Q(x)=m\right\}\) be its set of integral solutions for \(m\in{\mathbb Z}\). One is interested in the asymptotics of \(N(T)=\sharp\left\{x\in V_m({\mathbb Z}): \parallel x\parallel <T\right\}\).
Let \(\Gamma\subset SO(Q)^\circ\simeq SO(2,1)^\circ\) be an arithmetic lattice preserving \(V_m({\mathbb Z})\), then there are only finitely many \(\Gamma\)-orbits in \(V_m({\mathbb Z})\) and understanding the asymptotics of \(N(T)\) is reduced to understanding the asymptotics of \(\sharp(x\in v_0\Gamma:\parallel x\parallel <T)\) for \(v_0\in V_m({\mathbb Z})\).
Denoting by \(H\simeq SO(1,1)^\circ\) the identity component of the stabilizer of \(v_0\) in \(G\), one has \(vol(H\cap \Gamma/H)<\infty\) if and only if \(H\cap\Gamma\) is infinite. For this case the asymptotics of \(N_T\) has been determined by W. Duke et al. [Duke Math. J. 71, No. 1, 143–179 (1993; Zbl 0798.11024)] and by different methods by A. Eskin and C. McMullen [Duke Math. J. 71, No. 1, 181–209 (1993; Zbl 0798.11025)] and Y. Benoist and H. Oh [Ann. Inst. Fourier 62, No. 5, 1889–1942 (2012; Zbl 1330.11044)].
The paper under review presents the asymptotics (and the best possible error term) in the case of (not necessarily arithmetic) lattices with \(vol(H\cap \Gamma/H)=\infty\).
Several applications are discussed, such as to the limit distribution of orthogonal translates of a divergent geodesic, that is of \(\Gamma\backslash\Gamma Ha(T)\) in \(\Gamma\backslash{\mathbb H}^2\) for \(T\to\infty\) and \(a(T)=diag(e^{T/2},e^{-T/2})\), and to asymptotically counting integral binary quadratic forms with given discriminant and with coefficients bounded by \(T\).

MSC:

22E40 Discrete subgroups of Lie groups
11P21 Lattice points in specified regions
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References:

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