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On a theorem of Wirsing in Diophantine approximation


Author: Yann Bugeaud
Journal: Proc. Amer. Math. Soc. 144 (2016), 1905-1911
MSC (2010): Primary 11J04
DOI: https://doi.org/10.1090/proc/12879
Published electronically: October 2, 2015
MathSciNet review: 3460153
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Abstract: Let $ n$ and $ d$ be integers with $ 1 \le d \le n-1$. Let $ \xi $ be a real number which is not algebraic of degree at most $ n$. We establish that there exist an effectively computable constant $ c$, depending only on $ \xi $ and on $ n$, an integer $ k$ with $ 1 \le k \le d$, and infinitely many integer polynomials $ P(X)$ of degree $ m$ at most equal to $ n$ whose roots $ \alpha _1, \ldots , \alpha _m$ can be numbered in such a way that

$\displaystyle \vert(\xi - \alpha _1) \ldots (\xi - \alpha _k)\vert \le c \, H(P)^{-{d \over d+1}n - {1 \over d+1} - 1}. $

This extends a well-known result of Wirsing who dealt with the case $ d=1$.

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Additional Information

Yann Bugeaud
Affiliation: Mathématiques, Université de Strasbourg, 7, rue René Descartes, 67084 Strasbourg, France
Email: bugeaud@math.unistra.fr

DOI: https://doi.org/10.1090/proc/12879
Received by editor(s): March 13, 2015
Received by editor(s) in revised form: June 2, 2015
Published electronically: October 2, 2015
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2015 American Mathematical Society

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