EQUIDISTRIBUTION OF HIGHER DIMENSIONAL POLYNOMIAL TRAJECTORIES ON HOMOGENEOUS SPACES

The uploaded paper (Zhang, 2020) proves equidistribution for higher-dimensional regular algebraic trajectories Θ: R^k → G_1 averaged over anisotropically expanding boxes B_n = ∏[0, T_n^{λ_i}], under the hypothesis that the minimal closed subgroup F containing Θ(R^k) with FΓ closed is semisimple; see Theorem 1.2 and its reduction to Theorem 5.1 for the semisimple case, together with the program of non-divergence (Theorem 2.1), local unipotent invariance via twisting, linearization near singular sets, and induction on dim(G/Γ) . The candidate solution states the correct limiting measure and broadly cites the right rigidity inputs (Shah/Ratner), but it glosses over the central technical obstacle identified in the paper: for general higher-dimensional Θ (not of product type) one only gets local, not global, unipotent invariance without additional twisting and a careful analysis near singular sets; this is exactly why the paper develops Sections 3–4 before closing the argument in §5 . Moreover, the model implicitly assumes Γ is a lattice when asserting F ∩ gΓg^{-1} is a lattice in F (Step 1), whereas the paper explicitly treats Γ as a closed subgroup and uses reductions compatible with possible non-discreteness (Remark 1.5) . The model also jumps from some unipotent invariance to full F-invariance by invoking Borel–Tits, without addressing the need to construct enough unipotent invariance in the non-product, k ≥ 2 setting; the paper’s proof supplies that through twisting plus induction (Theorem 5.1) . Conclusion: the paper’s result and proof are sound; the model’s outline is incomplete and uses unjustified steps.