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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Cylindrical minima of integral lattices


Author: A. A. Illarionov
Translated by: S. V. Kislyakov
Original publication: Algebra i Analiz, tom 24 (2012), nomer 2.
Journal: St. Petersburg Math. J. 24 (2013), 301-312
MSC (2010): Primary 11H06
Published electronically: January 22, 2013
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Abstract: Let $ \Phi $ be a norm in $ \mathbb{R}^{s-1}$. A nonzero node $ \gamma = (\gamma _1,\dots ,\gamma _s)$ of an $ s$-dimensional lattice $ \Gamma $ is called a $ \Phi $-cylindrical minimum of $ \Gamma $ if there is no other nonzero node $ \eta = (\eta _1,\dots ,\eta _s)$ with

$\displaystyle \Phi (\gamma _1,\dots ,\gamma _{s-1}) \le \Phi (\eta _1,\dots ,\eta _{s-1}), \quad \vert\eta _s\vert \le \vert\gamma _s\vert, $

where at least one inequality is strict. It is proved that the average number of the $ \Phi $-cylindrical minima of $ s$-dimensional integer lattices whose determinant belongs to $ [1;N]$ is equal to $ \mathcal {C}_s(\Phi ) \cdot \ln N + O_{s,\Phi }(1)$, where $ \mathcal {C}_s(\Phi )$ is a positive constant depending only on $ s$ and $ \Phi $. This formula is a version of the classical result about the average length of a finite continued fraction.

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Additional Information

A. A. Illarionov
Affiliation: Institute of applied Mathematics, Russian Academy of Sciences, Far-East division, Dzerzhinskii st. 54, Khabarovsk 680000, Russia
Email: illar{\textunderscore}a@list.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-2013-01243-X
PII: S 1061-0022(2013)01243-X
Keywords: Lattice, relative minimum, multidimensional continued fraction, multidimensional best approximations
Received by editor(s): December 16, 2010
Published electronically: January 22, 2013
Additional Notes: The author was supported by RFBR (grants nos. 11-01-00628-a, 11-01-12004-ofi-m-2011), by the Presidium of the Far-East division of RAS (grant no. 12-I-II19-01), and by the grant NSh-1922.2012.1 for state support of leading scientific schools.
Article copyright: © Copyright 2013 American Mathematical Society