On uniform polynomial approximation

Poëls, Anthony

Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 769-807

Let $n$ be a positive integer and $\xi$ a transcendental real number. We are interested in bounding from above the uniform exponent of polynomial approximation ${\widehat{\omega }}_n\left(\xi \right)$. Davenport and Schmidt’s original 1969 inequality ${\widehat{\omega }}_n\left(\xi \right)\le 2n-1$ was improved recently, and the best upper bound known to date is $2n-2$ for each $n\ge 10$. In this paper, we develop new techniques leading us to the improved upper bound $2n-\frac{1}{3}{n}^{1/3}+𝒪\left(1\right)$.

Soient $n$ un entier strictement positif et $\xi$ un nombre réel transcendant. Nous cherchons à borner supérieurement l’exposant uniforme d’approximation polynomiale ${\widehat{\omega }}_n\left(\xi \right)$. Établie par Davenport et Schmidt en 1969, l’inégalité ${\widehat{\omega }}_n\left(\xi \right)\le 2n-1$, a été améliorée pour la première fois récemment, et la meilleure borne supérieure connue à ce jour est $2n-2$ pour tout $n\ge 10$. Dans ce papier, nous développons de nouvelles techniques qui nous permettent d’obtenir la borne supérieure améliorée $2n-\frac{1}{3}{n}^{1/3}+𝒪\left(1\right)$.

Classification : 11J13, 11J82
Keywords: Exponent of Diophantine approximation, heights, uniform polynomial approximation
Mots-clés : Exposants d’approximation diophantienne, hauteurs, approximation polynomiale uniforme

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