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Dal'nevost. Mat. Zh., 2011, Volume 11, Number 2, Pages 149–154 (Mi dvmg218)  

This article is cited in 1 scientific paper (total in 1 paper)

On the number of local minima of integer lattices

A. A. Illarionova, Y. A. Soykab

a Institute for Applied Mathematics, Khabarovsk Division, Far-Eastern Branch of the Russian Academy of Sciences, Khabarovsk
b Pacific National University, Khabarovsk

Abstract: Let $E_s(N)$ be the average number of local minima of $s$-dimensional integer lattices with determinant equals $N$. We prove the following estimates
$$ \frac{2^{-1}}{(s-1)!}+O_s(\frac{1}{\ln N})\le\frac{E_s(N)}{\ln^{s-1}N}\le\frac{2^s}{(s-1)!}+O_s(\frac{1}{\ln N}) $$
for any prime $N$. Using this result we have a new lower bound for maximum number of local minima of integer lattices.

Key words: local minimum, multidimensional continuous fraction.

Full text: PDF file (337 kB)
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Document Type: Article
UDC: 511.26, 511.9
MSC: Primary 11K60; Secondary 11G70
Received: 13.09.2011

Citation: A. A. Illarionov, Y. A. Soyka, “On the number of local minima of integer lattices”, Dal'nevost. Mat. Zh., 11:2 (2011), 149–154

Citation in format AMSBIB
\Bibitem{IllSoy11}
\by A.~A.~Illarionov, Y.~A.~Soyka
\paper On the number of local minima of integer lattices
\jour Dal'nevost. Mat. Zh.
\yr 2011
\vol 11
\issue 2
\pages 149--154
\mathnet{http://mi.mathnet.ru/dvmg218}


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    This publication is cited in the following articles:
    1. A. A. Illarionov, “A multidimensional generalization of Heilbronn's theorem on the average length of a finite continued fraction”, Sb. Math., 205:3 (2014), 419–431  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
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