Far Eastern Mathematical Journal

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On cylindrical minima of three-dimensional lattices


A. A. Illarionov

2011, issue 1, Ń. 37–47


Abstract
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
$$
\eta^2_1+\eta^2_2 \le \gamma^2_1+\gamma^2_2, \quad |\eta_3|\le |\gamma_3|, \quad |\gamma|<|\eta|.
$$
\noinden It is proved that the average number of cylindrical minima of three-dimensional integer lattices with determinant from $[1;N]$ is equal
$$
C?\ln N+O(1),
$$

Keywords:
minimum of lattice, multi-dimensional continuous fraction

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References

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