The Ramanujan Journal

, Volume 38, Issue 3, pp 513–528 | Cite as

A converse to linear independence criteria, valid almost everywhere

  • S. Fischler
  • M. Hussain
  • S. Kristensen
  • J. Levesley


We prove a weighted analogue of the Khintchine–Groshev theorem, where the distance to the nearest integer is replaced by the absolute value. This is applied to proving the optimality of several linear independence criteria over the field of rational numbers.

Mathematics Subject Classification

Primary 11J83 Secondary 11J72 11J13 



We are thankful to Yann Bugeaud and Detta Dickinson for suggesting us to work together, and to Michel Waldschmidt and an anonymous referee for their useful remarks.


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • S. Fischler
    • 1
  • M. Hussain
    • 2
  • S. Kristensen
    • 3
  • J. Levesley
    • 4
  1. 1.Equipe d’Arithmétique et de Géométrie Algébrique, Université Paris-SudOrsay CedexFrance
  2. 2.School of Mathematical and Physical SciencesThe University of NewcastleCallaghanAustralia
  3. 3.Department of MathematicsAarhus UniversityAarhus CDenmark
  4. 4.Department of MathematicsUniversity of YorkYorkUK

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