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
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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/

[1] Sam Chow; Anish Ghosh; Lifan Guan; Antoine Marnat; David Simmons Diophantine transference inequalities: weighted, inhomogeneous, and intermediate exponents (2019) (https://arxiv.org/abs/1808.07184v2, to appear in Ann. Sc. Norm. Super. Pisa, Cl. Sci.)

[2] Tushar Das; Lior Fishman; David Simmons; Mariusz Urbański A variational principle in the parametric geometry of numbers (2019) (https://arxiv.org/abs/1901.06602)

[3] Oleg German Transference theorems for Diophantine approximation with weights, Mathematika, Volume 66 (2020) no. 2, pp. 325-342 | Article | MR: 4130327 | Zbl: 1452.11080

[4] Antoine Marnat; Nikolay Moshchevitin An optimal bound for the ratio between ordinary and uniform exponents of Diophantine approximation, Mathematika, Volume 66 (2020) no. 3, pp. 818-854 | Article | MR: 4134247

[5] Martin Rivard-Cooke Parametric Geometry of Numbers (2019) (https://ruor.uottawa.ca/handle/10393/38871) (Ph. D. Thesis)

[6] Damien Roy On Schmidt and Summerer parametric geometry of numbers, Ann. Math., Volume 182 (2015) no. 2, pp. 739-786 | MR: 3418530 | Zbl: 1328.11076

[7] Damien Roy On the topology of Diophantine approximation spectra, Compos. Math., Volume 153 (2017) no. 7, pp. 1512-1546 | MR: 3705265 | Zbl: 1379.11069

[8] Wolfgang M. Schmidt On parametric geometry of numbers, Acta Arith., Volume 195 (2020) no. 4, pp. 383-414 | Article | MR: 4121876 | Zbl: 1452.11081

[9] Wolfgang M. Schmidt; Leonhard Summerer Parametric geometry of numbers and applications, Acta Arith., Volume 140 (2009) no. 1, pp. 67-91 | Article | MR: 2557854 | Zbl: 1236.11060

[10] Leonhard Summerer A geometric proof of Jarnik’s identity in the setting of weighted simultaneous approximation (2019) (https://arxiv.org/abs/1912.04574)

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