About Jarník’s-type relation in higher dimension
Annales de l'Institut Fourier, Volume 68 (2018) no. 1, pp. 131-150.

Using the Parametric Geometry of Numbers introduced recently by W. M. Schmidt and L. Summerer and results by D. Roy, we show that German’s transference inequalities between the two most classical exponents of uniform Diophantine approximation are optimal. Further, we establish that the n uniform exponents of Diophantine approximation in dimension n are algebraically independent. Thus, no Jarník’s-type relation holds between them.

En utilisant la géométrie paramétrique des nombres introduite récemment par W. M. Schmidt et L. Summerer et des résultats de D. Roy, nous montrons que les inégalités de transfert entre les deux exposants uniformes d’approximation diophantienne les plus classiques, établies par O. German, sont optimales. De plus, nous établissons que les n exposants d’approximation uniforme en dimension n sont algébriquement indépendants. Ainsi en dimension supérieure à 2, ils ne sont pas reliés par une relation de dépendance analogue à l’identité de Jarník.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3154
Classification: 11H06,  11J13
Keywords: Parametric geometry of numbers, Uniform exponents of Diophantine approximation, Transference inequalities.
Marnat, Antoine 1

1 Department of Mathematics University of York York YO10 5DD (UK)
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Marnat, Antoine. About Jarník’s-type relation in higher dimension. Annales de l'Institut Fourier, Volume 68 (2018) no. 1, pp. 131-150. doi : 10.5802/aif.3154. https://aif.centre-mersenne.org/articles/10.5802/aif.3154/

[1] Bugeaud, Yann; Laurent, Michel On exponents of homogeneous and inhomogeneous diophantine approximation, Mosc. Math. J., Volume 5 (2005), pp. 747-766

[2] Bugeaud, Yann; Laurent, Michel Exponents of Diophantine approximation, Diophantine Geometry Proceedings (Centro di Ricerca Matematica Ennio De Giorgi) Volume 4, Edizioni della Normale, 2007, pp. 101-121 | Zbl: 1229.11098

[3] Cheung, Yitwah Special divergent trajectories for a homogeneous flow (2008) (Seminar of Geometry, Chicago University, http://www.math.uchicago.edu/~geometry/gt_seminar.F2008.html)

[4] Cheung, Yitwah Prescriptions for a diagonal flow on the space of lattices. (2015) (Diophantine Approximation and Related Topics, Aarhus, http://mjcnt.phystech.edu/conference/aarhus/abstracts/Cheung.pdf)

[5] German, Oleg N. On Diophantine exponents and Khintchine’s transference principle, Mosc. J. Comb. Number Theory, Volume 2 (2012) no. 2, pp. 22-51 | MR: 2988525 | Zbl: 1294.11116

[6] Jarník, Vojtěch Zum Khintchineschen “Übertragungssatz”, Trav. Inst. Math. Tbilissi, Volume 3 (1938), pp. 193-212

[7] Khinchin, Alexander Ya. Über eine Klasse linearer diophantischer Approximationen, Rend. Circ. Mat. Palermo 50 (1926), pp. 170-195 | Article

[8] Khinchin, Alexander Ya. Zur metrischen Theorie der diophantischen Approximationen, Math.Z., Volume 24 (1926), pp. 706-714 | Article

[9] Khinchin, Alexander Ya. On some applications of the method of the additional variable, Am. Math. Soc., Transl., Volume 1950 (1950) no. 18, 14 pages | MR: 0035790 (12,12c)

[10] Laurent, Michel Exponents of Diophantine Approximmation in Dimension Two, Canad. J. Math., Volume 61 (2009), pp. 165-189 | Article

[11] Laurent, Michel On transfer inequalities in Diophantine approximation, Analytic number theory, Cambridge University Press, 2009, pp. 306-314 | MR: 2508652 (2010a:11132)

[12] Minkowski, Hermann Geometrie der Zahlen, Bibliotheca Mathematica Teubneriana, Band 40, Johnson Reprint Corp., New York-London, 1968, vii+256 pages | MR: 0249269 (40 #2515)

[13] Roy, Damien On Schmidt and Summerer parametric geometry of numbers, Ann. Math., Volume 182 (2015), pp. 739-786 | Article

[14] Roy, Damien Spectrum of the exponents of best rational approximation, Math. Z., Volume 283 (2016), pp. 143-155 | Article

[15] Schmidt, Wolfgang M. On heights of algebraic subspaces and diophantine approximations, Ann. Math., Volume 85 (1967), pp. 430-472 | Article | MR: 0213301 (35 #4165)

[16] Schmidt, Wolfgang M. Open problems in Diophantine approximation, Diophantine approximations and transcendental numbers (Luminy, 1982) (Progr. Math.) Volume 31, Birkhäuser Boston, Boston, MA, 1983, pp. 271-287 | MR: 702204

[17] Schmidt, Wolfgang M.; Summerer, Leonhard Parametric geometry of numbers and applications, Acta Arithmetica, Volume 140 (2009) no. 1, pp. 67-91 | Article

[18] Schmidt, Wolfgang M.; Summerer, Leonhard Diophantine approximation and parametric geometry of numbers, Monatsch. Math., Volume 169 (2013) no. 1, pp. 51-104 | Article

[19] Schmidt, Wolfgang M.; Summerer, Leonhard The generalization of Jarnik’s identity, Acta Arithmetica, Volume 175 (2016), pp. 119-136

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