Torus-packet equidistribution in prime, quartic, and sextic degrees

We prove the packet form of the higher-dimensional Duke equidistribution problem for totally real fields of any fixed prime degree at least five, allowing arbitrary local homothety types of full lattices. As the multiplier-order discriminant tends to infinity, the volume-weighted packet measures converge to Haar probability measure, with no escape of mass.

Let K range over totally real quartic fields with no proper intermediate field, and let \mathcal O be any order in K. We prove that the packets of periodic diagonal orbits attached to invertible \mathcal O-ideal classes, weighted by orbit volume, equidistribute with no escape of mass as |\mathop{Disc}\nolimits (\mathcal O)| tends to infinity. The proof combines measure rigidity with a cubic-resolvent estimate that remains uniform at primes dividing the order index.

We prove that the complete ideal-class packets of maximal orders in totally real sextic fields with no proper intermediate fields become equidistributed, with no escape of mass, in the space of unimodular lattices as their field discriminants tend to infinity. The limit is Haar probability measure. Each packet includes all coordinate-sign translates and is weighted by diagonal-orbit volume.