Completely empty pyramids on integer lattices and two-dimensional faces of multidimensional continued fractions

In this paper we develop an integer-affine classification of three-dimensional multistory, completely empty convex marked pyramids. We apply it to obtain the complete lists of compact two-dimensional faces of multidimensional continued fractions lying in planes at integer distances 2, 3, 4, …  to the origin. The faces are considered up to the action of the group of integer-linear transformations.     

Probability measures on [SIN] groups and some related ideals in group algebras

In 1921, Blichfeldt gave an upper bound on the number of integral points contained in a convex body in terms of the volume of the body. More precisely, he showed that , whenever is a convex body containing n + 1 affinely independent integral points. Here we prove an analogous inequality with respect to the surface area F(K), namely . The proof is based on a slight improvement of Blichfeldt’s bound in the case when K is a non-lattice translate of a lattice polytope, i.e., K = t + P, where and P is an n-dimensional polytope with integral vertices. Then we have . Moreover, in the 3-dimensional case we prove a stronger inequality, namely . Authors’ addresses: Martin Henk, Institut für Algebra und Geometrie, Universit?t Magdeburg, Universit?tsplatz 2, D-39106 Magdeburg, Germany; J?rg M. Wills, Mathematisches Institut, Universit?t Siegen, ENC, D-57068 Siegen, Germany     

Successive Minima and Best Simultaneous Diophantine Approximations

We study the problem of best approximations of a vector by rational vectors of a lattice whose common denominator is bounded. To this end we introduce successive minima for a periodic lattice structure and extend some classical results from geometry of numbers to this structure. This leads to bounds for the best approximation problem which generalize and improve former results.     

On the Density of 2-Saturated Lattice Packings of Discs

 In a recent paper of G. Fejes Tóth, G. Kuperberg and W. Kuperberg [1] a conjecture has been published concerning the greatest lower bound of the density of a 2-saturated packing of unit discs in the plane. (A packing of unit discs is said to be 2-saturated if none of the discs could be replaced by two other ones of the same size to generate a new packing. A packing of the unit disc is a lattice packing if the centers form a point lattice.) In the present note we study this problem for lattice packings, however, in a more general form in which the removed unit disc is replaced by two discs of radius r. A corollary of our results supports the above conjecture proving that a lattice packing cannot be 2-saturated except if its density is larger than the conjectured bound. (Received 6 December 2000; in revised form March 29, 2001)     

Remarks on my paper: packing of incongruent circles on the sphere

The paper [3] contains an upper bound to the weighted density of a packing of circles on the unit sphere with radii from a given finite set. This bound is attained by many packings and has applications to problems of solidity. In the present note it is shown that a certain condition imposed on the set of admissible radii can be removed by modifying the original proof of the theorem.     

Simultaneous Packing and Covering in the Euclidean Plane

 In this article we study the simultaneous packing and covering constants of two-dimensional centrally symmetric convex domains. Besides an identity result between translative case and lattice case and a general upper bound, exact values for some special domains are determined. Similar to Mahler and Reinhardt’s result about packing densities, we show that the simultaneous packing and covering constant of an octagon is larger than that of a circle. (Received 17 January 2001; in revised form 13 July 2001)     

Complementary Sets of Finite Sets

Subsets 𝒜, 𝒮 of an additive group G are complementary if 𝒜 + 𝒮 = G. When 𝒜 is of finite cardinality ∣𝒜∣, and G is ℤ or ℝ, we give sufficient conditions for the existence of a complementary set 𝒮 with “density” not much larger than 1/∣𝒜∣. Supported in part by NSF DMS-0074531. Received February 14, 2002; in revised form July 18, 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

Covering and packing in ${\Bbb Z}^n$ and ${\Bbb R}^n$, (I)

Given a finite subset of an additive group such as or , we are interested in efficient covering of by translates of , and efficient packing of translates of in . A set provides a covering if the translates with cover (i.e., their union is ), and the covering will be efficient if has small density in . On the other hand, a set will provide a packing if the translated sets with are mutually disjoint, and the packing is efficient if has large density. In the present part (I) we will derive some facts on these concepts when , and give estimates for the minimal covering densities and maximal packing densities of finite sets . In part (II) we will again deal with , and study the behaviour of such densities under linear transformations. In part (III) we will turn to . Authors’ address: Department of Mathematics, University of Colorado at Boulder, Campus Box 395, Boulder, Colorado 80309-0395, USA The first author was partially supported by NSF DMS 0074531.     

Symmetric Groups and Lattices

This paper deals with various problems in lattice theory involving local extrema. In particular, we construct infinite series of highly symmetric spherical 3-designs which include some of the examples constructed in [9] in dimensions 5 and 7. We also construct new types of dual-extreme lattices.Received June 29, 2002; in final form January 14, 2003 Published online May 16, 2003     

Counting Singular Matrices with Primitive Row Vectors

We solve an asymptotic problem in the geometry of numbers, where we count the number of singular n×n matrices where row vectors are primitive and of length at most T. Without the constraint of primitivity, the problem was solved by Y. Katznelson. We show that as T, the number is asymptotic to for n3. The 3-dimensional case is the most problematic and we need to invoke an equidistribution theorem due to W. M. Schmidt.     

Minkowski Arrangements of Spheres

Let be a non-negative number not greater than 1. Consider an arrangement of (not necessarily congruent) spheres with positive homogenity in the n-dimensional Euclidean space, i.e., in which the infimum of the radii of the spheres divided by the supremum of the radii of the spheres is a positive number. With each sphere S of associate a concentric sphere of radius times the radius of S. We call this sphere the -kernel of S. The arrangement is said to be a Minkowski arrangement of order if no sphere of overlaps the -kernel of another sphere. The problem is to find the greatest possible density of n-dimensional Minkowski sphere arrangements of order . In this paper we give upper bounds on for .     

A Property of Polynomials with an Applicationto Siegel’s Lemma

 It is proved that natural necessary conditions imply the existence of infinitely many integer points at which given multivariate polynomials with integer coefficients take coprime values. As a consequence the best constant in the simplest case of Siegel’s lemma is expressed in terms of critical determinants of suitable star bodies. Received August 10, 2001; in revised form March 13, 2002 RID="a" ID="a" Dedicated to Professor Edmund Hlawka on the occasion of his 85th birthday     

Homomorphisms on Function Lattices

In this paper we study real lattice homomorphisms on a unital vector lattice , where X is a completely regular space. We stress on topological properties of its structure spaces and on its representation as point evaluations. These results are applied to the lattice of real Lipschitz functions on a metric space. Using the automatic continuity of lattice homomorphisms with respect to the Lipschitz norm, we are able to derive a Banach-Stone theorem in this context. Namely, it is proved that the unital vector lattice structure of Lip (X) characterizes the Lipschitz structure of the complete metric space X. In the case of bounded Lipschitz functions, an analogous result is obtained in the class of complete quasiconvex metric spaces.     

On the Packing Densities and the Covering Densities of the Cartesian Products of Convex Bodies

In this article we study the following problem: Is the covering (packing) density of a Cartesian product of two convex bodies always equal to the product of their corresponding covering (packing) densities? For the covering case we get a negative answer. For the packing case we get a combinatorial version which seems to be important for its own interest.     

Etude du graphe divisoriel 2

 We show that the maximum number of positive integers that appear in k non disjoint paths of the divisorial graph restricted to the numbers ⩽N, is about N log k/log N. We study also some other related questions. Re?u le 20 juin 2000; en forme révisée le 4 Avril 2002     

The lattice discrepancy of bodies bounded by a rotating Lamé’s curv

This article provides an asymptotic result for the lattice point discrepancy of the special three-dimensional body for fixed k > 2 and large t. Authors’ addresses: Ekkehard Kr?tzel, Faculty of Mathematics, University of Vienna, Nordbergstra?e 15, 1090 Wien, ?sterreich; Werner Georg Nowak, Department of Integrative Biology, Institute of Mathematics, Universit?t für Bodenkultur Wien, Gregor Mendel-Stra?e 33, 1180 Wien, ?sterreich     

Eine explizite Abschätzung für die Gitter-Diskrepanz von Rotationsellipsoiden

An explicit estimate for the lattice point discrepancy of ellipsoids of rotation. For the lattice point discrepancy (i.e., the number of integer points minus the volume) of the ellipsoid (u ₁ ² + u ₂ ²)/a + a ² u ₃ ² ≤ x (a, x > 0), this paper provides an estimate of the form terms of smaller order in x. Die Autoren danken dem ?sterreichischen Fonds zur F?rderung der wissenschaftlichen Forschung (FWF) für finanzielle Unterstützung unter der Projekt-Nr. P18079-N12.     

The Best Constant in Siegel’s Lemma

We give a new proof of the basis form of Siegels Lemma over an algebraic number field k in which the field and dimension dependent constant is best possible. This constant is equal to a generalization of Hermites constant for the algebraic number field k that has recently been studied by J. L. Thunder.Research supported in part by the National Science Foundation (DMS-00-88915).Communicated by W. SchmidtReceived April 4, 2002; in revised form April 28, 2003 Published online August 28, 2003     

Algèbres de Lie résolubles réelles algébriquement rigides

Résumé.  We present all real solvable algebraically rigid Lie algebras of dimension lower or equal than eight. We point out the differences that distinguish the real and complex classification of solvable rigid Lie algebras.     

Lattice Points in Planar Convex Domains

We investigate the number of lattice points in planar convex domains. We give estimates of the remainder in the asymptotic representation with numerical constants, which are astonishingly small. We consider convex planar domains whose boundary has nonvanishing curvature throughout. Here the curvature of the curve of boundary plays an important role. Further, we consider the number of lattice points in domains which are bounded by superellipses. These curves have isolated points with curvature zero.