Geodesic
On cylindrical minima of three-dimensional lattices
Dalʹnevostočnyj matematičeskij žurnal, Tome 11 (2011) no. 1, pp. 37-47.

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Nonzero point $\gamma=(\gamma_1,\gamma_2,\gamma_3)$ of three-dimensional lattice $\Gamma$ is called by cylindrical minimum if there exist no nonzero point $\eta=(\eta_1,\eta_2,\eta_3)$ such as $$ \eta_1^2+\eta_2^2\le\gamma_1^2+\gamma_2^2, \quad |\eta_3|\le|\gamma_3|, \quad |\gamma||\eta|. $$ It is proved that the average number of cylindrical minima of three-dimensional integer lattices with determinant from $[1;N]$ is equal $$ \mathcal C \cdot\ln N+O(1). $$ 
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A. A. Illarionov. On cylindrical minima of three-dimensional lattices. Dalʹnevostočnyj matematičeskij žurnal, Tome 11 (2011) no. 1, pp. 37-47. http://geodesic.mathdoc.fr/item/DVMG_2011_11_1_a3/

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