The distribution of periodic torus orbits on homogeneous spaces

TL;DR

This paper investigates the distribution of periodic torus orbits on homogeneous spaces, proving results that support standard conjectures up to small exceptional sets, and applies these findings to refine Minkowski's theorem on ideal classes.

Contribution

It establishes new results on the equidistribution of periodic torus orbits, including bounds on exceptional sets, and connects these to number theory applications.

Findings

Certain conjectures hold up to O(Δ^ε) orbits of discriminant ≤ Δ.

Examples show some orbits do not become equidistributed.

Application to sharpening Minkowski's theorem on ideal classes.

Abstract

We prove results towards the equidistribution of certain families of periodic torus orbits on homogeneous spaces, with particular focus on the case of the diagonal torus acting on quotients of $\PGL_n(\R)$. After attaching to each periodic orbit an integral invariant (the discriminant) our results have the following flavour: certain standard conjectures about the distribution of such orbits hold up to exceptional sets of at most $O(\Delta ^{\epsilon})$ orbits of discriminant $\leq \Delta$. The proof relies on the well-separatedness of periodic orbits together with measure rigidity for torus actions. We also give examples of sequences of periodic orbits of this action that fail to become equidistributed, even in higher rank. We give an application of our results to sharpen a theorem of Minkowski on ideal classes in totally real number fields of cubic and higher degrees.