On Kronecker's density theorem, primitive points and orbits of matrices

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We discuss recent quantitative results in connexion with Kronecker's theorem on the density of subgroups in Rⁿ and with Dani and Raghavan's theorem on the density of orbits in the spaces of frames. We also propose several related problems. The case of the natural linear action of the unimodular group SL₂(Z) on the real plane is investigated more closely. We then establish an intriguing link between the configuration of (discrete) orbits of primitive points and the rate of density of dense orbits.