Geodesic
On the number of local minima of integer lattices
Dalʹnevostočnyj matematičeskij žurnal, Tome 11 (2011) no. 2, pp. 149-154.

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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 $$ \frac{2^{-1}}{(s-1)!}+O_s\left(\frac{1}{\ln N}\right)\le\frac{E_s(N)}{\ln^{s-1}N}\le\frac{2^s}{(s-1)!}+O_s\left(\frac{1}{\ln N}\right) $$ for any prime $N$. Using this result we have a new lower bound for maximum number of local minima of integer lattices. 
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A. A. Illarionov; Y. A. Soyka. On the number of local minima of integer lattices. Dalʹnevostočnyj matematičeskij žurnal, Tome 11 (2011) no. 2, pp. 149-154. http://geodesic.mathdoc.fr/item/DVMG_2011_11_2_a2/

[1] G. F. Voronoi, Sobranie sochinenii v 3-kh tomakh, v. 1, AN USSR, Kiev, 1952

[2] H. Minkowski, “Generalisation de la theorie des fraction continues”, Ann. Sci. École Norm. Sup., 13:2 (1896), 41–60 | MR | Zbl

[3] V. A. Bykovskii, “O pogreshnosti teoretiko-chislovykh kvadraturnykh formul”, DAN, 389:2 (2003), 154–155 | MR | Zbl

[4] N. M. Korobov, Teoretiko-chislovye metody v priblizhennom analize, MTsNMO, M., 2004 | MR

[5] M. O. Avdeeva, “Otsenka kolichestva lokalnykh minimumov tselochislennykh reshetok”, Chebyshevskii sbornik, 5:4 (2004), 35–38 | MR | Zbl

[6] O. A. Gorkusha, N. M. Dobrovolskii, “Ob otsenkakh giperbolicheskoi dzeta-funktsii reshetok”, Chebyshevskii sbornik, 6:2 (2005) | MR | Zbl

[7] M. O. Avdeeva, “O nizhnikh otsenkakh kolichestva lokalnykh minimumov tselochislennykh reshetok”, Fundamentalnaya i prikladnaya matematika, 11:6 (2005), 9–14 | MR

[8] M. O. Avdeeva, V. A. Bykovskii, “Verkhnie i nizhnie otsenki konstanty Voronogo-Minkovskogo”, Matem. zametki, 87:4 (2010), 483–491 | DOI | MR | Zbl

[9] H. Heilbronn, “On the average length of a class of finite continued fractions”, Abhandlungen aus zahlentheorie und Analysis, VEB, Berlin, 1968, 89–96 | MR

[10] J. W. Porter, “On a theorem of Heilbronn”, Mathematika, 22:1 (1975), 20–28 | DOI | MR | Zbl

[11] A. A. Illarionov, Srednee kolichestvo otnositelnykh minimumov trekhmernykh tselochislennykh reshetok fiksirovannogo opredelitelya, Preprint KhO IMP DVO RAN No 2, DalNauka, Vladivostok, 2010

[12] A. A. Illarionov, “Multidimentional generalisation of Heilbronn's theorem about average length of finite continued fractions”, Abstracts of 27th Journees Arithmetiques, 2011

[13] Dzh. V. Kassels, Vvedenie v geometriyu chisel, Mir, M., 1995