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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Simultaneous rational approximation
to binomial functions


Author: Michael A. Bennett
Journal: Trans. Amer. Math. Soc. 348 (1996), 1717-1738
MSC (1991): Primary 11J68, 11J82; Secondary 11D57
MathSciNet review: 1321566
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Abstract: We apply Padé approximation techniques to deduce lower bounds for simultaneous rational approximation to one or more algebraic numbers. In particular, we strengthen work of Osgood, Fel´dman and Rickert, proving, for example, that

\begin{displaymath}\max \left\{ \left| \sqrt{2} - p_{1}/q \right| , \left| \sqrt{3} - p_{2}/q \right| \right\} > q^{-1.79155} \end{displaymath}

for $q > q_{0}$ (where the latter is an effective constant). Some of the Diophantine consequences of such bounds will be discussed, specifically in the direction of solving simultaneous Pell's equations and norm form equations.


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Additional Information

Michael A. Bennett
Affiliation: Department of Pure Mathematics, The University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1
Address at time of publication: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: mabennet@math.lsa.umich.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01480-8
PII: S 0002-9947(96)01480-8
Keywords: Simultaneous approximation to algebraic numbers, irrationality and linear independence measures, Padé approximants, Pell-type equations, norm form equations
Received by editor(s): June 30, 1994
Received by editor(s) in revised form: January 31, 1995
Additional Notes: Research supported by an NSERC Postdoctoral Fellowship
Article copyright: © Copyright 1996 American Mathematical Society