Orbit distribution on \(\mathbb R^2\) under the natural action of \(\mathrm{SL}(2,\mathbb Z)\)
The distribution of orbits under the natural linear action of \(\mathrm{SL}(2,\mathbb Z\) on the plane is studied. An application/example is the evaluation of the asymptotic behaviour of the set \(\{A\in \mathrm{SL}(2,\mathbb Z)\mid Ax\in [0,r]^2, \|A\|\leq t\}\) for large \(r,t>0\), where \(x\) is an explicit vector with incommensurable entries. More general results in this direction have been obtained by \textit{F. Ledrappier} [C. R. Acad. Sci., Paris, Sér. I, Math. 329, No. 1, 61--64 (1999; Zbl 0928.22012)] and others using the ergodic theory of Lie group actions. In this paper more restricted results are reached using number theory methods, giving sharper and more explicit error terms.
scientific article; zbMATH DE number 1893633