Extremum properties of lattice packing and covering with circles.

*(English)*Zbl 1329.05061Let \(L\) be a lattice in the plane and write \(\delta(L)\) and \(\nu(L)\) for the packing and covering density of \(L\), respectively. Further, let \(\delta^*(L)\) and \(\nu^*(L)\) be the packing and covering densities of the polar lattice \(L^*\). The paper is devoted to a uniform study of local maximum properties of functions \(\delta\), \(\delta \delta^*\), \(\delta \nu\) and local minimum properties of functions \(\nu\), \(\nu \nu^*\), \(\nu/\delta\). Due to the fact that the polar of a planar lattice is essentially a rotated copy of it, the results for \(\delta \delta^*\) and \(\nu \nu^*\) follow from those for \(\delta\) and \(\nu\).

The paper contains a variety of results in this direction. To describe their general flavor, we need some notation. Let \(0\) and \(I\) be the \(2 \times 2\) zero and identity matrices, respectively, and let \(A = (a_{ik})\) be a \(2 \times 2\) symmetric matrix with trace equal to zero. Let \[ \| A\| = \left( \sum_{i=1}^2 \sum_{k=1}^2 a_{ik}^2 \right)^{1/2} \] be the \(\ell_2\)-norm of \(A\). The local behavior of the density function \(\delta\) can be analyzed by looking at the quotient \(\delta((I+A)L)/\delta(L)\) as \(A \to 0\). The author distinguishes four situations:

The paper contains a variety of results in this direction. To describe their general flavor, we need some notation. Let \(0\) and \(I\) be the \(2 \times 2\) zero and identity matrices, respectively, and let \(A = (a_{ik})\) be a \(2 \times 2\) symmetric matrix with trace equal to zero. Let \[ \| A\| = \left( \sum_{i=1}^2 \sum_{k=1}^2 a_{ik}^2 \right)^{1/2} \] be the \(\ell_2\)-norm of \(A\). The local behavior of the density function \(\delta\) can be analyzed by looking at the quotient \(\delta((I+A)L)/\delta(L)\) as \(A \to 0\). The author distinguishes four situations:

- {\(\bullet\)}
- if the quotient is \(\leq 1 + o(\| A\|)\), \(\delta\) is upper semi-stationary at \(L\);
- {\(\bullet\)}
- if the quotient is \(= 1 + o(\| A\|)\), \(\delta\) is stationary at \(L\);
- {\(\bullet\)}
- if the quotient is \(\leq 1\), \(\delta\) is maximum at \(L\);
- {\(\bullet\)}
- if the quotient is \(\leq 1 - \mathrm{const }\| A\|\), \(\delta\) is ultra-maximum at \(L\).

Reviewer: Lenny Fukshansky (Claremont)

##### MSC:

05B40 | Combinatorial aspects of packing and covering |

11H06 | Lattices and convex bodies (number-theoretic aspects) |

11H31 | Lattice packing and covering (number-theoretic aspects) |

11H50 | Minima of forms |

52C15 | Packing and covering in \(2\) dimensions (aspects of discrete geometry) |

52C17 | Packing and covering in \(n\) dimensions (aspects of discrete geometry) |