On uniform polynomial approximation

Anthony Poëls

doi:10.5802/jep.265

On uniform polynomial approximation
[Sur l’approximation polynomiale uniforme]

Anthony Poëls ¹

¹ Universite Claude Bernard Lyon 1, Institut Camille Jordan UMR 5208, 69622 Villeurbanne, France

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

Résumé
Abstract

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

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

Reçu le : 2024-05-16
Accepté le : 2024-07-18
Publié le : 2024-07-23

DOI : 10.5802/jep.265

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

Affiliations des auteurs :

Anthony Poëls ¹

¹ Universite Claude Bernard Lyon 1, Institut Camille Jordan UMR 5208, 69622 Villeurbanne, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Anthony Poëls. On uniform polynomial approximation. Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 769-807. doi : 10.5802/jep.265. https://jep.centre-mersenne.org/articles/10.5802/jep.265/

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