Cylindrical minima of integral lattices

Abstract:

Let $\Phi$ be a norm in $\mathbb {R}^{s-1}$. A nonzero node $\gamma = (\gamma _1,\dots ,\gamma _s)$ of an $s$-dimensional lattice $\Gamma$ is called a $\Phi$-cylindrical minimum of $\Gamma$ if there is no other nonzero node $\eta = (\eta _1,\dots ,\eta _s)$ with \[ \Phi (\gamma _1,\dots ,\gamma _{s-1}) \le \Phi (\eta _1,\dots ,\eta _{s-1}), \quad |\eta _s| \le |\gamma _s|, \] where at least one inequality is strict. It is proved that the average number of the $\Phi$-cylindrical minima of $s$-dimensional integer lattices whose determinant belongs to $[1;N]$ is equal to $\mathcal {C}_s(\Phi ) \cdot \ln N + O_{s,\Phi }(1)$, where $\mathcal {C}_s(\Phi )$ is a positive constant depending only on $s$ and $\Phi$. This formula is a version of the classical result about the average length of a finite continued fraction.