Geodesic
Distribution of facets of higher-dimensional Klein polyhedra
Sbornik. Mathematics, Tome 209 (2018) no. 1, pp. 56-70

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We consider facets of Klein polyhedra of a given integer-linear type $\mathscr T$ in a certain lattice. Let $E_\mathscr T(N,s)$ be the typical number of facets, averaged over all integral $s$-dimensional lattices with determinant $N$. Assume that the interior of any facet of type $\mathscr T$ contains at least one point of the corresponding lattice. We prove that $$ E_\mathscr T(N,s)=C_\mathscr T \ln^{s-1}N+O_\mathscr T (\ln^{s-2} N \cdot \ln\ln N) \quad\text{as } N \to \infty, $$ where $C_\mathscr T$ is a positive constant depending only on $\mathscr T$. Bibliography: 28 titles.
Keywords: lattice, Klein polyhedron, multidimensional continued fraction. 
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     author = {A. A. Illarionov},
     title = {Distribution of facets of higher-dimensional {Klein} polyhedra},
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     url = {http://geodesic.mathdoc.fr/item/SM_2018_209_1_a2/}
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A. A. Illarionov. Distribution of facets of higher-dimensional Klein polyhedra. Sbornik. Mathematics, Tome 209 (2018) no. 1, pp. 56-70. http://geodesic.mathdoc.fr/item/SM_2018_209_1_a2/