A transference principle for simultaneous rational approximation
Journal de Théorie des Nombres de Bordeaux, Tome 32 (2020) no. 2, pp. 387-402.

Nous établissons pour tout entier n1 un principe de transfert général concernant la mesure d’irrationalité des points de n+1 dont les coordonnées sont linéairement indépendantes sur . 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 -linearly independent coordinates in n+1 , for any given integer n1. 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.

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DOI : https://doi.org/10.5802/jtnb.1127
Classification : 11J13,  11J82
Mots clés : exponents of Diophantine approximation, heights, Marnat–Moshchevitin transference inequalities, measures of rational approximation, simultaneous approximation
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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/

[1] Vojtěch Jarník Zum Khintchineschen “Übertragungssatz”, Tr. Tbilis. Mat. Inst., Volume 3 (1938), pp. 193-212 | Zbl 0019.10602

[2] Aleksandr Khintchine Über eine Klasse linearer diophantischer Approximationen, Rend. Circ. Mat. Palermo, Volume 50 (1926), pp. 170-195 | Article | Zbl 52.0183.01

[3] Aleksandr Khintchine Zur metrischen Theorie der diophantischen Approximationen, Math. Z., Volume 24 (1926) no. 1, pp. 706-714 | Article | MR 1544787 | Zbl 52.0183.02

[4] Dmitry Kleinbock; Nikolay Moshchevitin Simultaneous Diophantine approximation: sums of squares and homogeneous polynomials, Acta Arith., Volume 190 (2019) no. 1, pp. 87-100 | Article | MR 3976428 | Zbl 07087815

[5] Michel Laurent Exponents of Diophantine approximation in dimension two, Can. J. Math., Volume 61 (2009) no. 1, pp. 165-189 | Article | MR 2488454 | Zbl 1229.11101

[6] Antoine Marnat; Nikolay Moshchevitin An optimal bound for the ratio between ordinary and uniform exponents of Diophantine approximation, Mathematika, Volume 66 (2020) no. 3, pp. 818-854 | Article | MR 4134247

[7] Nikolay Moshchevitin Exponents for three-dimensional simultaneous Diophantine approximations, Czech. Math. J., Volume 62 (2012) no. 1, pp. 127-137 | Article | MR 2899740 | Zbl 1249.11061

[8] Ngoc Ai Van Nguyen On some problems in Transcendental number theory and Diophantine approximation (2014) (https://ruor.uottawa.ca/handle/10393/30350) (Ph. D. Thesis)

[9] Anthony Poëls; Damien Roy Rational approximation to real points on quadratic hypersurfaces (2019), 25 pages (https://arxiv.org/abs/1909.01499, to appear in J. Lond. Math. Soc. (2))

[10] Martin Rivard-Cooke Parametric Geometry of Numbers (2019) (https://ruor.uottawa.ca/handle/10393/38871) (Ph. D. Thesis)

[11] Damien Roy Approximation to real numbers by cubic algebraic integers I, Proc. Lond. Math. Soc., Volume 88 (2004) no. 1, pp. 42-62 | MR 2018957 | Zbl 1035.11028

[12] Damien Roy Rational approximation to real points on conics, Ann. Inst. Fourier, Volume 63 (2013) no. 6, pp. 2331-2348 | Numdam | MR 3237450 | Zbl 1304.11064

[13] Wolfgang M. Schmidt Diophantine Approximation, Lecture Notes in Mathematics, Volume 785, Springer, 1980 | Zbl 0421.10019

[14] Wolfgang M. Schmidt Diophantine Approximations and Diophantine Equations, Lecture Notes in Mathematics, Volume 1467, Springer, 1991 | MR 1176315 | Zbl 0754.11020

[15] Wolfgang M. Schmidt; Leonhard Summerer Simultaneous approximation to three numbers, Mosc. J. Comb. Number Theory, Volume 3 (2013) no. 1, pp. 84-107 | MR 3284111 | Zbl 1301.11058

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