On consecutive Farey arcs
On Convex Bodies that Permit Packings of High Density.
On coverings with convex domains
On extreme forms in dimension 8
A theorem of Voronoi asserts that a lattice is extreme if and only if it is perfect and eutactic. Very recently the classification of the perfect forms in dimension has been completed [5]. There are 10916 perfect lattices. Using methods of linear programming, we are able to identify those that are additionally eutactic. In lower dimensions almost all perfect lattices are also eutactic (for example out of the in dimension ). This is no longer the case in dimension : up to similarity, there...
On Gauss's proof of Seeber's Theorem.
On geodesics of phyllotaxis
Seeds of sunflowers are often modelled by leading to a roughly uniform repartition with seeds indexed by consecutive integers at angular distance for the golden ratio. We associate to such a map a geodesic path of the modular curve and use it for local descriptions of the image of the phyllotactic map .
On integer points in polygons
The phenomenon of anomaly small error terms in the lattice point problem is considered in detail in two dimensions. For irrational polygons the errors are expressed in terms of diophantine properties of the side slopes. As a result, for the -dilatation, , of certain classes of irrational polygons the error terms are bounded as with some , or as with arbitrarily small .
On lattice bases with special properties
In this paper we introduce multiplicative lattices in and determine finite unions of suitable simplices as fundamental domains for sublattices of finite index. For this we define cyclic non-negative bases in arbitrary lattices. These bases are then used to calculate Shintani cones in totally real algebraic number fields. We mainly concentrate our considerations to lattices in two and three dimensions corresponding to cubic and quartic fields.
On Lattice Points in Polyhedral Cross-Sections.
On lattice polytopes having interior lattice points.
On Linear Sections of Lattice Packings.
On lower bounds of the density of Delone sets and holes in sequences of sphere packings.
On Norm Three Vectors in Integral Euclidean Lattices. I.
On perfect forms in six variables.
On Siegel's Lemma.
On systems of linear inequalities
We show in detail that the category of general Roth systems or the category of semi-stable systems of linear inequalities of slope zero is a neutral Tannakian category. On the way, we present a new proof of the semi-stability of the tensor product of semi-stable systems. The proof is based on a numerical criterion for a system of linear inequalities to be semi-stable.
On ternary cubic forms.
On the computation of Hermite-Humbert constants for real quadratic number fields
We present algorithms for the computation of extreme binary Humbert forms in real quadratic number fields. With these algorithms we are able to compute extreme Humbert forms for the number fields and . Finally we compute the Hermite-Humbert constant for the number field .
On the construction of dense lattices with a given automorphisms group
We consider the problem of constructing dense lattices in with a given non trivial automorphisms group. We exhibit a family of such lattices of density at least , which matches, up to a multiplicative constant, the best known density of a lattice packing. For an infinite sequence of dimensions , we exhibit a finite set of lattices that come with an automorphisms group of size , and a constant proportion of which achieves the aforementioned lower bound on the largest packing density. The algorithmic...
On the Covering Multiplicity of Lattices.