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Sbornik: Mathematics, 2018, Volume 209, Issue 1, Pages 56–70
DOI: https://doi.org/10.1070/SM8788
(Mi sm8788)
 

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
References:
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.
Funding agency Grant number
Russian Science Foundation 14-11-00335
This research was supported by the Russian Science Foundation (project no. 14-11-00335).
Received: 15.07.2016 and 19.04.2017
Russian version:
Matematicheskii Sbornik, 2018, Volume 209, Number 1, Pages 58–73
DOI: https://doi.org/10.4213/sm8788
Bibliographic databases:
Document Type: Article
UDC: 511.36+511.9
Language: English
Original paper language: Russian
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
Citation in format AMSBIB
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\paper Distribution of facets of higher-dimensional Klein polyhedra
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Linking options:
  • https://www.mathnet.ru/eng/sm8788
  • https://doi.org/10.1070/SM8788
  • https://www.mathnet.ru/eng/sm/v209/i1/p58
  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Математический сборник Sbornik: Mathematics
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    Abstract page:523
    Russian version PDF:54
    English version PDF:17
    References:52
    First page:19
     
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