A graph arising in the Geometry of Numbers
Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 251-260.

The parametric geometry of numbers has allowed to visualize the simultaneous approximation properties of a collection of real numbers through the combined graph of the related successive minima functions. Several inequalities among classical exponents of simultaneous approximation can be guessed by a study of these graphs; in particular the so called regular graph is of major importance as it provides an extremal case for some of these inequalities. The aim of this paper is to define and construct an analogue of the regular graph in the case of weighted simultaneous approximation.

La géometrie paramétrique des nombres a permis de visualiser les propriétés d’approximation simultanée d’une collection de nombres réels à travers le graphe combiné des fonctions de certains minimas successifs. Beaucoup d’inégalités entre les exposants classiques d’approximation simultanée peuvent être déduits de ces graphes. En particulier, les graphes dits réguliers sont parmis les plus importants, notamment pour les cas extrêmes de certaines de ces inégalités. Le but de cet article est de définir et de construire la notion de graphes réguliers dans le contexte d’approximation pondérée.

Published online:
DOI: 10.5802/jtnb.1160
Classification: 11H06, 11J13
Keywords: Parametric Geometry of numbers, successive minima, simultaneous approximation
Wolfgang M. Schmidt 1; Leonhard Summerer 2

1 Department of Mathematics University of Colorado Boulder, CO 80309-0395, USA
2 Fakultät für Mathematik Universität Wien Oskar-Morgenstern-Platz 1 A-1090 Wien, Austria
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Wolfgang M. Schmidt; Leonhard Summerer. A graph arising in the Geometry of Numbers. Journal de théorie des nombres de Bordeaux, Volume 33 (2021) no. 1, pp. 251-260. doi : 10.5802/jtnb.1160. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1160/

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