Independence of the Diophantine exponents associated with linear subspaces

We elaborate on a problem raised by Schmidt in 1967 which generalizes the theory of classical Diophantine approximation to subspaces of ${\mathbb{R}}^n$. We consider Diophantine exponents for linear subspaces of ${\mathbb{R}}^n$ which generalize the irrationality measure for real numbers. We prove here that we have no smooth relations among some functions associated to these exponents. To establish this result, we construct subspaces for which we are able to compute the exponents.