Arithmetic Quantum Unique Ergodicity for Products of Hyperbolic 2- and 3-Manifolds

The paper proves AQUE for congruence quotients of S=(H(2))^r×(H(3))^s (Theorem 1) by the standard five-step strategy: microlocal lifting to X=Γ\G with AM-invariant limits, positive-entropy for A1, measure classification to homogeneous components, a novel amplifier/induction to rule out mass on xHM-type sets, and a separate no-escape-of-mass input. These elements are explicitly laid out and proved, including the statement of Theorem 1 and the new Theorem 2 plus the amplifier framework and induction step (e.g., the sketch and inequalities (1.2)–(1.3), and the culminating Theorem 35) . The microlocal lift and A1-invariance are stated precisely in the homogeneity section, and positive-entropy bounds for tubes are established (Proposition 13) . Measure rigidity is invoked via a rank-one, positive-entropy classification (Theorem 17), giving homogeneous components supported on closed H-orbits that contain a real-rank-1 subgroup at the first factor; the remaining possibility is eliminated by the new amplifier/induction method . No-escape-of-mass is addressed as a separate input (Zaman, extending Soundararajan), as summarized in Step (5) of the paper’s strategy . The candidate solution mirrors this structure closely and correctly. Minor imprecisions include citing EKL06 (higher-rank) instead of the rank-one EL08 result used here, and conflating A with A1, but these do not affect the logical core. Overall, both are correct and follow substantially the same proof.