Quadratic forms of signature (2, 2) or (3, 1) I: effective equidistribution in quotients of SL4(R)

Zuo Lin

correctmedium confidence

Category: Not specified
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:57 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper’s Theorem 1.1 states exactly the dichotomy the model proposes—polynomial-rate equidistribution of the U-box average after a_log T or proximity to a periodic H-orbit with controlled volume—specialized to H ≅ SO(2,2)^0 or SO(3,1)^0 embedded in SL4(R) . The paper achieves this via a three-phase method (effective closing + dimension improvement via sheeted sets/Margulis functions + mixing) , whereas the model outlines a classical Dani–Margulis linearization + Eskin–Margulis–Mozes approach with exponential mixing and quantitative non-divergence. The results align, but the proof strategies are substantially different.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

This paper establishes a polynomially effective equidistribution theorem for horospherical orbits of SO(2,2)° and SO(3,1)° inside SL4(R)/Γ, extending the frontier of quantitative homogeneous dynamics beyond the quasi-split, absolute rank-2 settings. The argument synthesizes an effective closing lemma, an avoidance principle, and a sophisticated dimension-improvement scheme on the Ad(H)-invariant complement, culminating in a clean mixing step. The results are important for effective Oppenheim-type problems. Some expository refinements would further enhance readability.