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

Victor Beresnevich, D. Kleinbock, G. Margulis

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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.
Original languageEnglish
Pages (from-to)1-31
Number of pages31
JournalJournal de Théorie des Nombres de Bordeaux
Issue number1
Early online date21 May 2015
Publication statusE-pub ahead of print - 21 May 2015

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  • metric simultaneous Diophantine approximation
  • linear forms
  • strongly extremal manifolds
  • multiplicatively very well approximable points

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