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JOURNALS // Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya // Archive

Izv. RAN. Ser. Mat., 2012 Volume 76, Issue 3, Pages 111–138 (Mi im6035)

This article is cited in 6 papers

The average number of relative minima of three-dimensional integer lattices of a given determinant

A. A. Illarionov

Institute for Applied Mathematics, Khabarovsk Division, Far-Eastern Branch of the Russian Academy of Sciences

Abstract: We obtain an asymptotic formula for the average number of relative minima of the three-dimensional complete integer lattices of a given determinant. This generalizes Heilbronn's classical result on the average length of a finite continued fraction with a fixed denominator.

Keywords: relative minimum, multidimensional continued fraction, average length of continued fractions.

UDC: 511.36+511.9

MSC: Primary 11H06; Secondary 11H56, 11H60

Received: 05.11.2010

DOI: 10.4213/im6035


 English version:
Izvestiya: Mathematics, 2012, 76:3, 535–562

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© Steklov Math. Inst. of RAS, 2025