About Jarník’s-type relation in higher dimension

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 are algebraically independent. Thus, no Jarník’s-type relation holds between them.