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Simultaneous rational approximation to binomial functions

Authors:	Michael A Bennett

Institution:	Department of Pure Mathematics, The University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1

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.

Keywords:	Simultaneous approximation to algebraic numbers irrationality and linear independence measures Padé approximants Pell-type equations norm form equations

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