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March 2019 On the discrepancy between best and uniform approximation
Johannes Schleischitz
Funct. Approx. Comment. Math. 60(1): 21-29 (March 2019). DOI: 10.7169/facm/1642

Abstract

For $\zeta$ a transcendental real number, we consider the classical Diophantine exponents $w_{n}(\zeta)$ and $\widehat{w}_{n}(\zeta)$. They measure how small $\vert P(\zeta)\vert$ can be for an integer polynomial $P$ of degree at most $n$ and naive height bounded by $X$, for arbitrarily large and all large $X$, respectively. The discrepancy between the exponents $w_{n}(\zeta)$ and $\widehat{w}_{n}(\zeta)$ has attracted interest recently. Studying parametric geometry of numbers, W. Schmidt and L. Summerer were the first to refine the trivial inequality $w_{n}(\zeta)\geq \widehat{w}_{n}(\zeta)$. Y. Bugeaud and the author found another estimation provided that the condition $w_{n}(\zeta)>w_{n-1}(\zeta)$ holds. In this paper we establish an unconditional version of the latter result, which can be regarded as a proper extension. Unfortunately, the new contribution involves an additional exponent and is of interest only in certain cases.

Citation

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Johannes Schleischitz. "On the discrepancy between best and uniform approximation." Funct. Approx. Comment. Math. 60 (1) 21 - 29, March 2019. https://doi.org/10.7169/facm/1642

Information

Published: March 2019
First available in Project Euclid: 28 March 2018

zbMATH: 07055561
MathSciNet: MR3932601
Digital Object Identifier: 10.7169/facm/1642

Subjects:
Primary: 11J13
Secondary: 11J25 , 11J82

Keywords: diophantine inequalities , exponents of Diophantine approximation , U-numbers

Rights: Copyright © 2019 Adam Mickiewicz University

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Vol.60 • No. 1 • March 2019
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