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Uniform Diophantine approximation and best approximation polynomials

Volume 185 / 2018

Johannes Schleischitz Acta Arithmetica 185 (2018), 249-274 MSC: Primary 11J13; Secondary 11H06. DOI: 10.4064/aa170901-4-7 Published online: 3 August 2018


Let $\zeta$ be a transcendental real number. We introduce a new method to find upper bounds for the classical exponent $\widehat{w}_{n}(\zeta)$ concerning uniform polynomial approximation. Our method is based on the parametric geometry of numbers introduced by Schmidt and Summerer, and transference of the original approximation problem in dimension $n$ to suitable higher dimensions. For large $n$, we can provide an unconditional bound of order $\widehat{w}_{n}(\zeta)\leq 2n-2+o(1)$. While this improves the bound of order $2n-{3/2}+o(1)$ due to Bugeaud and the author, it is unfortunately slightly weaker than what can be obtained when incorporating a recently proved conjecture of Schmidt and Summerer. However, the method also enables us to establish a significantly stronger conditional bound upon a certain presumably weak assumption on the structure of the best approximation polynomials. Thereby we provide strong evidence that the known upper bounds for the exponent are crude.


  • Johannes SchleischitzDepartment of Mathematics and Statistics
    University of Ottawa
    King Edward 585
    Ottawa, ON, Canada K1N 6N5

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