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We describe new arithmetic invariants for pairs of torus orbits on groupsisogenous to an inner form of $\mathbf{PGL}_n$ over a number field. Theseinvariants are constructed by studying the double quotient of a linearalgebraic group by a maximal torus.  Using the new invariants we significantly strengthen results towards theequidistribution of packets of periodic torus orbits on higher rank$S$-arithmetic quotients. Packets of periodic torus orbits are naturalcollections of torus orbits coming from a single adelic torus and are closelyrelated to class groups of number fields. The distribution of these orbits isakin to the distribution of integral points on homogeneous algebraic varietieswith a torus stabilizer.  The proof combines geometric invariant theory, Galois actions, localarithmetic estimates and homogeneous dynamics.

Arithmetic of Double Torus Quotients and the Distribution of Periodic Torus Orbits