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
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
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