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A lattice in Euclidean $d$-space is called well-rounded if it contains $d$ linearly independent vectors of minimal length. This class of lattices is important for various questions, including sphere packing or homology computations. The task of enumerating well-rounded sublattices of a given lattice is of interest already in dimension 2, and has recently been treated by several authors. In this paper, we analyse the question more closely in the spirit of earlier work on similar sublattices and coincidence site sublattices. Combining explicit geometric considerations with known techniques from the theory of Dirichlet series, we arrive, after a considerable amount of computation, at asymptotic results on the number of well-rounded sublattices up to a given index in any planar lattice. For the two most symmetric lattices, the square and the hexagonal lattice, we present detailed results.
The correspondence between unmixed bipartite graphs and sublattices of the oolean lattice is discussed. By using this correspondence, we show the existence of squarefree quadratic initial ideals of toric ideals arising from minimal vertex covers of unmixed bipartite graphs.
For a positive integer $s$, a lattice $L$ is said to be $s$-integrable if $sqrt{s}cdot L$ is isometric to a sublattice of $mathbb{Z}^n$ for some integer $n$. Conway and Sloane found two minimal non $2$-integrable lattices of rank $12$ and determinant
New series of $2^{2m}$-dimensional universally strongly perfect lattices $Lambda_I $ and $Gamma_J $ are constructed with $$2BW_{2m} ^{#} subseteq Gamma _J subseteq BW_{2m} subseteq Lambda _I subseteq BW _{2m}^{#} .$$ The lattices are found by restric
In this short note, we reformulate the task of calculating the pair correlation statistics of a Kronecker sequence as a lattice point counting problem. This can be done analogously to the lattice based approach which was used to (re-)prove the famous
We classify the dual strongly perfect lattices in dimension 16. There are four pairs of such lattices, the famous Barnes-Wall lattice $Lambda _{16}$, the extremal 5-modular lattice $N_{16}$, the odd Barnes-Wall lattice $O_{16}$ and its dual, and one