About Jarník’s-type relation in higher dimension

Marnat, Antoine

Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 131-150.

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.

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.

Classification : 11H06, 11J13
Keywords: Parametric geometry of numbers, Uniform exponents of Diophantine approximation, Transference inequalities.
Mot clés : Geometry paramétrique des nombres, Exposants d’approximation diophantienne uniformes, Inégalités de transfert.

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