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

TY - JOUR

T1 - Non-planarity and metric Diophantine approximation for systems of linear forms

AU - Beresnevich, Victor

AU - Kleinbock, D.

AU - Margulis, G.

N1 - © 2015, Publisher. This is an author-produced version of a paper accepted for publication. Uploaded with permission of the publisher/copyright holder. Further copying may not be permitted; contact the publisher for details

PY - 2015/5/21

Y1 - 2015/5/21

N2 - 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 of mxn matrices over R is introduced and studied. This notion generalises the one of non-planarity in R^n and is used to establish strong (Diophantine) extremality of manifolds and measures. The notion of weak non-planarity is shown to be `near optimal' in a certain sense. 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 due to Beresnevich and Velani and use them to bring the inhomogeneous theory in balance with its homogeneous counterpart.

AB - 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 of mxn matrices over R is introduced and studied. This notion generalises the one of non-planarity in R^n and is used to establish strong (Diophantine) extremality of manifolds and measures. The notion of weak non-planarity is shown to be `near optimal' in a certain sense. 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 due to Beresnevich and Velani and use them to bring the inhomogeneous theory in balance with its homogeneous counterpart.

KW - metric simultaneous Diophantine approximation

KW - linear forms

KW - strongly extremal manifolds

KW - multiplicatively very well approximable points

U2 - 10.5802/jtnb.890

DO - 10.5802/jtnb.890

M3 - Article

SN - 1246-7405

VL - 27

SP - 1

EP - 31

JO - Journal de Théorie des Nombres de Bordeaux

JF - Journal de Théorie des Nombres de Bordeaux

IS - 1

ER -