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On a generalization of Craig lattices

Hao Chen (2013)

Journal de Théorie des Nombres de Bordeaux

In this paper we introduce generalized Craig lattices, which allows us to construct lattices in Euclidean spaces of many dimensions in the range 3332 - 4096 which are denser than the densest known Mordell-Weil lattices. Moreover we prove that if there were some nice linear binary codes we could construct lattices even denser in the range 128 - 3272 . We also construct some dense lattices of dimensions in the range 4098 - 8232 . Finally we also obtain some new lattices of moderate dimensions such as 68 , 84 , 85 , 86 , which are denser than the...

On a generalization of the Selection Theorem of Mahler

Gilbert Muraz, Jean-Louis Verger-Gaugry (2005)

Journal de Théorie des Nombres de Bordeaux

The set 𝒰 𝒟 r of point sets of n , n 1 , having the property that their minimal interpoint distance is greater than a given strictly positive constant r > 0 is shown to be equippable by a metric for which it is a compact topological space and such that the Hausdorff metric on the subset 𝒰 𝒟 r , f 𝒰 𝒟 r of the finite point sets is compatible with the restriction of this topology to 𝒰 𝒟 r , f . We show that its subsets of Delone sets of given constants in n , n 1 , are compact. Three (classes of) metrics, whose one of crystallographic nature,...

On a question of Schmidt and Summerer concerning 3 -systems

Johannes Schleischitz (2020)

Communications in Mathematics

Following a suggestion of W.M. Schmidt and L. Summerer, we construct a proper 3 -system ( P 1 , P 2 , P 3 ) with the property ϕ ¯ 3 = 1 . In fact, our method generalizes to provide n -systems with ϕ ¯ n = 1 , for arbitrary n 3 . We visualize our constructions with graphics. We further present explicit examples of numbers ξ 1 , ... , ξ n - 1 that induce the n -systems in question.

On computations with integer division

Bettina Just, Friedhelm Meyer auf der Heide, Avi Wigderson (1989)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

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