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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Higher diophantine approximation exponents and continued fraction symmetries for function fields II
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by Dinesh S. Thakur PDF
Proc. Amer. Math. Soc. 141 (2013), 2603-2608 Request permission

Abstract:

We construct families of non-quadratic algebraic laurent series (over finite fields of any characteristic) which have only bad rational approximations so that their rational approximation exponent is as near to 2 as we wish and, at the same time, have very good quadratic approximations so that the quadratic exponent is close to the Liouville bound and thus can be arbitrarily large. In contrast, in the number field case, the Schmidt exponent (an analog of the Roth exponent of 2 for rational approximation) for approximations by quadratics is $3$. We do this by exploiting the symmetries of the relevant continued fractions. We then generalize some of the aspects from the degree $2 (=p^0+1)$-approximation to degree $p^n+1$-approximation. We also calculate the rational approximation exponent of an analog of $\pi$.
References
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Additional Information
  • Dinesh S. Thakur
  • Affiliation: Department of Mathematics, University of Arizona, P. O. Box 210089, Tucson, Arizona 85721-0089
  • Email: thakur@math.arizona.edu
  • Received by editor(s): August 13, 2011
  • Received by editor(s) in revised form: September 1, 2011, and November 5, 2011
  • Published electronically: April 17, 2013
  • Additional Notes: This work was supported in part by NSA grant H98230-10-1-0200
  • Communicated by: Matthew A. Papanikolas
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 141 (2013), 2603-2608
  • MSC (2010): Primary 11J68, 11J70, 11J93
  • DOI: https://doi.org/10.1090/S0002-9939-2013-11538-8
  • MathSciNet review: 3056550