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This article is cited in 1 scientific paper (total in 1 paper)
Distribution of facets of higher-dimensional Klein polyhedra
A. A. Illarionov Khabarovsk Division of the Institute for Applied Mathematics, Far Eastern Branch, Russian Academy of Sciences
Abstract:
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.
Received: 15.07.2016 and 19.04.2017
Citation:
A. A. Illarionov, “Distribution of facets of higher-dimensional Klein polyhedra”, Mat. Sb., 209:1 (2018), 58–73; Sb. Math., 209:1 (2018), 56–70
Linking options:
https://www.mathnet.ru/eng/sm8788https://doi.org/10.1070/SM8788 https://www.mathnet.ru/eng/sm/v209/i1/p58
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Abstract page: | 523 | Russian version PDF: | 54 | English version PDF: | 17 | References: | 52 | First page: | 19 |
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