POLYNOMIAL MIXING FOR TIME-CHANGES OF UNIPOTENT FLOWS

The paper proves that admissible smooth time-changes of unipotent flows on finite-volume homogeneous spaces have polynomial decay of correlations, under a strong spectral gap, via an L2-based mixing-via-shearing argument that controls ergodic integrals on large-measure sets and estimates shear/distortion of short geodesic arcs (Theorem 5; Definition 3; Lemmas 7–10; Proposition 13; end-of-proof) . The candidate solution reaches the same conclusion and follows the same high-level scheme (spectral-gap-driven bounds, shearing, integration by parts) but diverges technically in two ways: (i) it asserts an exact conjugacy identity for time-changed flows with a wrong e^s factor (the paper only uses the unperturbed commutation ht ◦ φ^X_r = φ^X_r ◦ h_{e^r t} and then develops a cocycle-based distortion analysis) ; and (ii) it invokes exponential mixing for the φ^X flow to average along geodesic arcs, which is stronger than what the paper needs (the paper uses an L2 integration-by-parts estimate instead) . With those corrections/clarifications, the candidate’s proof is consistent with the paper’s result.