The distribution of intersections in ${\mathrm{SL}(3, \mathbb{Z}) \backslash \mathrm{SL}(3, \mathbb{R})}$ and lattices related to roots of cubic congruences

In this note we study the distribution of the intersections between certain translates of closed orbits of the positive diagonal subgroup in $\SL(3, \mathbb{Z}) \backslash \SL(3, \mathbb{R})$ with a maximal parabolic subgroup. These intersections are closely connected to roots of congruences for certain monic, irreducible cubic polynomials $F(X) \in \mathbb{Z}[X]$. The main result is that the intersections, considered as a sequences in the diagonal subgroup and the parabolic subgroup, are jointly equidistributed. This implies that certain affine lattices determined by pairs of roots of the cubic congruences are jointly equidistributed with corresponding ideals in the associated ring of integers. We note that the techniques here roughly parallel those which have been developed to study the multidimensional Farey sequence, and one hopes that techniques to study roots of congruences will continue to develop.