Simultaneous rational approximation to binomial functions

Bennett, Michael

doi:10.1090/S0002-9947-96-01480-8

Simultaneous rational approximation to binomial functions
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by Michael A. Bennett

Trans. Amer. Math. Soc. 348 (1996), 1717-1738

DOI: https://doi.org/10.1090/S0002-9947-96-01480-8

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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 \[ \max \left \{ \left | \sqrt {2} - p_{1}/q \right | , \left | \sqrt {3} - p_{2}/q \right | \right \} > q^{-1.79155} \] 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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