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Mat. Sb., 2018, Volume 209, Number 1, Pages 58–73 (Mi msb8788)  

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
$$ E_\mathscr T(N,s)=C_\mathscr T \ln^{s-1}N+O_\mathscr T (\ln^{s-2} N \cdot \ln\ln N) \quadas N \to \infty, $$
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).


DOI: https://doi.org/10.4213/sm8788

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English version:
Sbornik: Mathematics, 2018, 209:1, 56–70

Bibliographic databases:

Document Type: Article
UDC: 511.36+511.9
MSC: 11J70, 11K50, 11K60, 11H06
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

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