Multiplicative zero-one laws and metric number theory

Volume 160 / 2013

Victor Beresnevich, Alan Haynes, Sanju Velani Acta Arithmetica 160 (2013), 101-114 MSC: 11J13, 11J83, 11K60. DOI: 10.4064/aa160-2-1

Abstract

We develop the classical theory of Diophantine approximation without assuming monotonicity or convexity. A complete `multiplicative' zero-one law is established akin to the `simultaneous' zero-one laws of Cassels and Gallagher. As a consequence we are able to establish the analogue of the Duffin–Schaeffer theorem within the multiplicative setup. The key ingredient is the rather simple but nevertheless versatile `cross fibering principle'. In a nutshell it enables us to `lift' zero-one laws to higher dimensions.

Authors

  • Victor BeresnevichDepartment of Mathematics
    University of York
    Heslington, York, YO10 5DD, England
    e-mail
  • Alan HaynesDepartment of Mathematics
    University of York
    Heslington, York, YO10 5DD, England
    e-mail
  • Sanju VelaniDepartment of Mathematics
    University of York
    Heslington, York, YO10 5DD, England
    e-mail

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