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

TL;DR

This paper explores recent quantitative results on the density of subgroups and orbits in Euclidean spaces, focusing on Kronecker's theorem, primitive points, and the action of SL_2(Z), revealing links between orbit configurations and density rates.

Contribution

It provides new insights into the density properties of orbits under linear group actions and connects orbit configurations with their density rates.

Findings

Quantitative results on subgroup density in R^n.

Analysis of orbit density under SL_2(Z) action.

Link between primitive point orbits and density rates.

Abstract

We discuss recent quantitative results in connexion with Kronecker's theorem on the density of subgroups in R^n 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 SL_2(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.