zbMATH — the first resource for mathematics

Extremum properties of lattice packing and covering with circles. (English) Zbl 1329.05061
Let \(L\) 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:
if the quotient is \(\leq 1 + o(\| A\|)\), \(\delta\) is upper semi-stationary at \(L\);
if the quotient is \(= 1 + o(\| A\|)\), \(\delta\) is stationary at \(L\);
if the quotient is \(\leq 1\), \(\delta\) is maximum at \(L\);
if the quotient is \(\leq 1 - \mathrm{const }\| A\|\), \(\delta\) is ultra-maximum at \(L\).
Here we say that the inequality holds as \(A \to 0\) if it holds for all \(A\) for which \(\| A\|\) is sufficiently small and \(\mathrm{const}\) is a positive constant. The local notions of upper semi-stationarity, stationarity, extremality and ultra-extremality can also be analogously defined for the other density functions and products under consideration. The author reviews a variety of known results (often with new proofs), as well as obtains some new ones, on lattices at which these properties are assumed; for example, not surprisingly \(\delta\) is upper semi-stationary if and only if \(L\) is the square or the hexagonal lattice. The author remarks that the strong property of ultra-extremality seems to be new, and proves that (more surprisingly) the packing density maximizers are also ultra-maximizers. Further, he shows that there are some rather unsymmetric parallelogram lattices for which \(\delta \nu\) is upper semi-stationary. All in all, the paper is full of classical and new observations, confirming the special status of the hexagonal lattice, but also uncovering some unexpected properties of less remarkable lattices.
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)
Full Text: DOI