NON-EXPANDING RANDOM WALKS ON HOMOGENEOUS SPACES AND DIOPHANTINE APPROXIMATION

Gaurav Aggarwal, Anish Ghosh

correctmedium confidence

Category: Not specified
Journal tier: Strong Field
Processed: Sep 28, 2025, 12:56 AM
arXiv Links: Abstract ↗PDF ↗

Audit review

The paper proves Theorem 1.1 via conditional measures along the horocycle-like R^{m+n}-action, an exponential drift mechanism (Prop. 6.1), and a translated-homogeneous toral decomposition culminating in Proposition 8.3, ruling out proper stabilizers and forcing full translation-invariance, hence µ=µ_X when π_*µ=µ_{X'} . The candidate solution’s Step 1 misuses the semidirect identity (it needs A_g^{-1}, not A_g) and applies a uniform W_1 bound that fails on the noncompact quotient X; it would also (incorrectly) force U-invariance in the purely linear case, contradicting the paper’s remark that non-virtual-linearity is necessary . Therefore the paper’s argument is sound, while the model’s proof contains critical errors.

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The paper introduces a robust framework to handle non-expanding random walks, a regime not covered by prior expanding-based techniques. The exponential drift and the translated-homogeneous toral decomposition are both original and effective. While the arguments are technically involved, they are coherent and appear correct. Minor expository improvements could broaden accessibility and highlight the conceptual innovations.