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A converse to linear independence criteria, valid almost everywhere

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Abstract

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.

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Acknowledgments

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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Authors and Affiliations

  1. Equipe d’Arithmétique et de Géométrie Algébrique, Université Paris-Sud, Bâtiment 425, 91405, Orsay Cedex, France

    S. Fischler

  2. School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, 2308, NSW, Australia

    M. Hussain

  3. Department of Mathematics, Aarhus University, Ny Munkegade 118, 8000, Aarhus C, Denmark

    S. Kristensen

  4. Department of Mathematics, University of York, York, YO10 5DD, UK

    J. Levesley

Authors
  1. S. Fischler
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  2. M. Hussain
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  3. S. Kristensen
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  4. J. Levesley
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Corresponding author

Correspondence to M. Hussain.

Additional information

S. Kristensen’s research was supported by the Danish Research Council for Independent Research, M. Hussain’s research was supported by the Australian Research Council, and S. Fischler’s research was partially supported by Agence Nationale de la Recherche (Project HAMOT, ref. ANR 2010 BLAN-0115-01).

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Fischler, S., Hussain, M., Kristensen, S. et al. A converse to linear independence criteria, valid almost everywhere. Ramanujan J 38, 513–528 (2015). https://doi.org/10.1007/s11139-014-9662-8

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  • DOI: https://doi.org/10.1007/s11139-014-9662-8

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