Non-planarity and metric Diophantine approximation for systems of linear forms

Victor Beresnevich; Dmitry Kleinbock; Gregory Margulis

doi:10.5802/jtnb.890

Victor Beresnevich ¹; Dmitry Kleinbock ²; Gregory Margulis ³

¹ University of York Heslington York, YO10 5DD, UK
² Brandeis University Waltham MA 02454-9110, USA
³ Yale University New Haven, CT 06520, USA

Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 1, pp. 1-31.

Abstract
Résumé

In this paper we develop a general theory of metric Diophantine approximation for systems of linear forms. A new notion of ‘weak non-planarity’ of manifolds and more generally measures on the space $M_{m, n}$ of $m \times n$ matrices over $ℝ$ is introduced and studied. This notion generalizes the one of non-planarity in $ℝ^{n}$ and is used to establish strong (Diophantine) extremality of manifolds and measures in $M_{m, n}$ . Thus our results contribute to resolving a problem stated in [20, §9.1] regarding the strong extremality of manifolds in $M_{m, n}$ . Beyond the above main theme of the paper, we also develop a corresponding theory of inhomogeneous and weighted Diophantine approximation. In particular, we extend the recent inhomogeneous transference results of the first named author and Velani [11] and use them to bring the inhomogeneous theory in balance with its homogeneous counterpart.

Dans cet article, nous développons la théorie métrique générale des approximations diophantiennes pour les systèmes de formes linéaires. Nous introduisons et puis étudions une nouvelle notion de « non-planéité faible » des variétés, et plus généralement des mesures sur l’espace $M_{m, n}$ des matrices $m \times n$ avec coefficients dans $ℝ$ . Cette notion généralise celle de non-planéité dans $ℝ^{n}$ . Nous utilisons cette notion pour établir une extrémalité forte (au sens diophantien) des variétés et des mesures de $M_{m, n}$ . Ainsi, nos résultats contribuent à la résolution d’un problème mentionné dans [20, §9.1] concernant l’extrémalité forte des variétés dans $M_{m, n}$ . Outre ce thème principal, nous développons aussi la théorie inhomogène et la théorie des approximations diophantiennes pondérées. En particulier, nous étendons les résultats récents sur le principe de transfert inhomogène du premier auteur et de Velani [11] et utilisons ce nouveau résultat pour mettre la théorie inhomogène en équilibre avec son homologue homogène.

MR: 3346961 | Zbl: 06554394

DOI: 10.5802/jtnb.890

Classification: 11J83, 11J13, 11K60

Author's affiliations:

Victor Beresnevich ¹; Dmitry Kleinbock ²; Gregory Margulis ³

¹ University of York Heslington York, YO10 5DD, UK
² Brandeis University Waltham MA 02454-9110, USA
³ Yale University New Haven, CT 06520, USA

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     author = {Victor Beresnevich and Dmitry Kleinbock and Gregory Margulis},
     title = {Non-planarity and metric {Diophantine} approximation for systems of linear forms},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {1--31},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {27},
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     year = {2015},
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     mrnumber = {3346961},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.890/}
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Victor Beresnevich; Dmitry Kleinbock; Gregory Margulis. Non-planarity and metric Diophantine approximation for systems of linear forms. Journal de théorie des nombres de Bordeaux, Volume 27 (2015) no. 1, pp. 1-31. doi : 10.5802/jtnb.890. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.890/

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