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 | References | Similar Articles | Additional Information

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 . We do this by exploiting the symmetries of the relevant continued fractions. We then generalize some of the aspects from the degree -approximation to degree -approximation. We also calculate the rational approximation exponent of an analog of .

**[BS76]**Leonard E. Baum and Melvin M. Sweet,*Continued fractions of algebraic power series in characteristic 2*, Ann. of Math. (2)**103**(1976), no. 3, 593–610. MR**0409372****[BG06]**Enrico Bombieri and Walter Gubler,*Heights in Diophantine geometry*, New Mathematical Monographs, vol. 4, Cambridge University Press, Cambridge, 2006. MR**2216774****[B04]**Yann Bugeaud,*Approximation by algebraic numbers*, Cambridge Tracts in Mathematics, vol. 160, Cambridge University Press, Cambridge, 2004. MR**2136100****[KTV00]**Minhyong Kim, Dinesh S. Thakur, and José Felipe Voloch,*Diophantine approximation and deformation*, Bull. Soc. Math. France**128**(2000), no. 4, 585–598 (English, with English and French summaries). MR**1815399****[M49]**K. Mahler,*On a theorem of Liouville in fields of positive characteristic*, Canadian J. Math.**1**(1949), 397–400. MR**0031497****[dM70]**Bernard de Mathan,*Approximations diophantiennes dans un corps local*, Bull. Soc. Math. France Suppl. Mém.**21**(1970), 93 (French). MR**0274396****[S80]**Wolfgang M. Schmidt,*Diophantine approximation*, Lecture Notes in Mathematics, vol. 785, Springer, Berlin, 1980. MR**568710****[S00]**Wolfgang M. Schmidt,*On continued fractions and Diophantine approximation in power series fields*, Acta Arith.**95**(2000), no. 2, 139–166. MR**1785412****[T99]**Dinesh S. Thakur,*Diophantine approximation exponents and continued fractions for algebraic power series*, J. Number Theory**79**(1999), no. 2, 284–291. MR**1728151**, 10.1006/jnth.1999.2413**[T04]**Dinesh S. Thakur,*Function field arithmetic*, World Scientific Publishing Co., Inc., River Edge, NJ, 2004. MR**2091265****[T09]**Dinesh S. Thakur,*Approximation exponents for function fields*, Analytic number theory, Cambridge Univ. Press, Cambridge, 2009, pp. 421–435. MR**2508661****[T11]**Dinesh S. Thakur,*Higher Diophantine approximation exponents and continued fraction symmetries for function fields*, Proc. Amer. Math. Soc.**139**(2011), no. 1, 11–19. MR**2729066**, 10.1090/S0002-9939-2010-10584-1**[V88]**J. F. Voloch,*Diophantine approximation in positive characteristic*, Period. Math. Hungar.**19**(1988), no. 3, 217–225. MR**961018**, 10.1007/BF01850290**[W]**M. Waldschmidt,*Report on some recent advances in Diophantine approximation*, to be published (Springer) in Special volume in honor of Serge Lang. Also at people.math.jussieu.fr/miw/articles/pdf/miwLangMemorialVolume.pdf

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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

DOI:
https://doi.org/10.1090/S0002-9939-2013-11538-8

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.