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Abstract |
Dirichlet's theorem in Diophantine approximation implies that for any real x, there exists a rational p/q arbitrarily close to x such that $|x-p/q| < 1/q^2$. In addition, the exponent 2 that appears in this inequality is optimal, as seen for example by taking $x=\sqrt2$. In 1967, Wolfgang Schmidt suggested a similar problem, where x is a real subspace of $R^d$ of dimension $\ell$, which one seeks to approximate by a rational subspace v. Our goal will be to obtain the optimal value of the exponent in the analogue of Dirichlet's theorem within this framework. The proof is based on a study of diagonal orbits in the space of lattices in $R^d$. |
