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Higher diophantine approximation exponents and continued fraction symmetries for function fields II

Author: Dinesh S. Thakur
Journal: Proc. Amer. Math. Soc. 141 (2013), 2603-2608
MSC (2010): Primary 11J68, 11J70, 11J93
Published electronically: April 17, 2013
MathSciNet review: 3056550
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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 $.

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Additional Information

Dinesh S. Thakur
Affiliation: Department of Mathematics, University of Arizona, P. O. Box 210089, Tucson, Arizona 85721-0089

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
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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