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Average number of local minima for three-dimensional integral lattices


Author: A. A. Illarionov
Translated by: S. Kislyakov
Original publication: Algebra i Analiz, tom 23 (2011), nomer 3.
Journal: St. Petersburg Math. J. 23 (2012), 551-570
MSC (2010): Primary 11H06
DOI: https://doi.org/10.1090/S1061-0022-2012-01208-2
Published electronically: March 2, 2012
MathSciNet review: 2896168
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Abstract: An asymptotic formula is found for the average number of local minima of three-dimensional complete integral lattices with determinant in the interval $ [1,N]$. This is a generalization to the two-dimensional case of the classical result about the average length of a finite continued fraction with denominator belonging to $ [1,N]$.


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Additional Information

A. A. Illarionov
Affiliation: Khabarovsk Division, Institute of Applied Mathematics, Russian Academy of Sciences, 54 Dzerzhinskiǐ Street, Khabarovsk 680000, Russia
Email: illar_a@list.ru

DOI: https://doi.org/10.1090/S1061-0022-2012-01208-2
Keywords: Lattice, local minimum, multidimensional continued fraction, Euclid algorithm
Received by editor(s): November 30, 2009
Published electronically: March 2, 2012
Additional Notes: Supported by RFBR (grants nos. 10-01-98002r-siberia-a , 11-01-00628-a), by FED RAS (grants nos. 11-III-V-01M-002, 09-I-114-03), and by the grant MD-2339.2010.1 of the President of RF
Article copyright: © Copyright 2012 American Mathematical Society