A transference principle for simultaneous rational approximation

Nguyen, Ngoc Ai Van; Poëls, Anthony; Roy, Damien

Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 2, pp. 387-402.

Nous établissons pour tout entier $n\ge 1$ un principe de transfert général concernant la mesure d’irrationalité des points de ${ℝ}^{n+1}$ dont les coordonnées sont linéairement indépendantes sur $\mathbb{Q}$. 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.

We establish a general transference principle about the irrationality measure of points with $\mathbb{Q}$-linearly independent coordinates in ${ℝ}^{n+1}$, for any given integer $n\ge 1$. 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.

Classification : 11J13, 11J82
Mots clés : exponents of Diophantine approximation, heights, Marnat–Moshchevitin transference inequalities, measures of rational approximation, simultaneous approximation

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