Higher diophantine approximation exponents and continued fraction symmetries for function fields

Author:
Dinesh S. Thakur

Journal:
Proc. Amer. Math. Soc. **139** (2011), 11-19

MSC (2010):
Primary 11J68, 11J70, 11J93

DOI:
https://doi.org/10.1090/S0002-9939-2010-10584-1

Published electronically:
August 18, 2010

MathSciNet review:
2729066

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Abstract: We construct many families of nonquadratic algebraic Laurent series with continued fractions having a bounded partial quotients sequence (the diophantine approximation exponent for approximation by rationals is thus , agreeing with the Roth value) and with the diophantine approximation exponent for approximation by quadratics being arbitrarily large. In contrast, the Schmidt value (analog of the Roth value for approximations by quadratics, in the number field case) is . We calculate diophantine approximation exponents for approximations by rationals for function field analogs of , and Hurwitz numbers (which are transcendental) and also give an interesting lower bound (which may be the actual value) for the exponent for approximation by quadratics for the latter two. We do this exploiting the situation when `folding' or `negative reversal' patterns of the relevant continued fractions become `repeating' or `half-repeating' in even or odd characteristic respectively.

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Additional Information

**Dinesh S. Thakur**

Affiliation:
Department of Mathematics, University of Arizona, 617 N. Santa Rita Avenue, Tucson, Arizona 85721

Email:
thakur@@math.arizona.edu

DOI:
https://doi.org/10.1090/S0002-9939-2010-10584-1

Received by editor(s):
January 2, 2010

Published electronically:
August 18, 2010

Additional Notes:
This research was supported in part by NSA grant H98230-08-1-0049

Communicated by:
Matthew A. Papanikolas

Article copyright:
© Copyright 2010
Dinesh S. Thakur