A transference principle for simultaneous rational approximation
Ngoc Ai Van Nguyen1 ; Anthony Poëls2 ; Damien Roy2
1 University of Information Technology Vietnam National University, Ho Chi Minh City, Vietnam
2 Département de Mathématiques Université d’Ottawa 150 Louis Pasteur Ottawa, Ontario K1N 6N5, Canada
Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 2, pp. 387-402.

We establish a general transference principle about the irrationality measure of points with ℚ-linearly independent coordinates in ℝn+1, for any given integer n≥1. On this basis, we recover an important inequality of Marnat and Moshchevitin which describes the spectrum of the pairs of ordinary and uniform exponents of rational approximation to those points. For points whose pair of exponents are close to the boundary in the sense that they almost realize the equality, we provide additional information about the corresponding sequence of best rational approximations. We conclude with an application.

Nous établissons pour tout entier n≥1 un principe de transfert général concernant la mesure d’irrationalité des points de ℝn+1 dont les coordonnées sont linéairement indépendantes sur ℚ. Partant de là nous retrouvons une inégalité importante de Marnat et Moshchevitin qui décrit le spectre conjoint des exposants ordinaire et uniforme d’approximation rationnelle pour ces points. Lorsque les exposants d’un point réalisent quasiment l’égalité, nous fournissons davantage d’informations sur la suite de ses meilleures approximations rationnelles. Nous concluons avec une application.

Reçu le :
Accepté le :
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DOI : 10.5802/jtnb.1127
Classification : 11J13, 11J82
Keywords: exponents of Diophantine approximation, heights, Marnat–Moshchevitin transference inequalities, measures of rational approximation, simultaneous approximation

Ngoc Ai Van Nguyen 1 ; Anthony Poëls 2 ; Damien Roy 2

1 University of Information Technology Vietnam National University, Ho Chi Minh City, Vietnam
2 Département de Mathématiques Université d’Ottawa 150 Louis Pasteur Ottawa, Ontario K1N 6N5, Canada
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Ngoc Ai Van Nguyen; Anthony Poëls; Damien Roy. A transference principle for simultaneous rational approximation. Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 2, pp. 387-402. doi : 10.5802/jtnb.1127. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1127/

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