Equidistribution of polynomially bounded o-minimal curves in homogeneous spaces

Michael Bersudsky, Nimish A. Shah, Hao Xing

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 equidistribution for non-contracting o-minimal definable curves via: (i) establishing the hull H_ϕ and a bounded correcting curve β with β(t) → β∞ (Theorem 2.5), (ii) (C,α)-goodness for definable, polynomially bounded families to get non-escape (Section 4), (iii) unipotent invariance (Section 5), and (iv) avoidance of singular sets plus Ratner to identify the unique homogeneous limit (Section 6), yielding Theorem 1.7. The candidate solution follows the same architecture and ingredients. See Theorem 1.7 and its surrounding discussion for the statement and correction-by-β limit β∞−1μLx (), the hull/correcting curve construction (), (C,α)-goodness and non-escape (; ), and the unipotent invariance/avoidance arguments (; ; conclusion ).

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The manuscript proves an equidistribution theorem for non-contracting curves definable in polynomially bounded o-minimal structures, extending Shah’s polynomial-curve result and integrating Ratner theory with a new (C,α)-goodness theorem for definable families. The strategy—hull and correction, non-escape, unipotent invariance, linearization with avoidance of singular sets, and Ratner classification—is standard but adapted with substantial new technical input. The arguments appear correct and complete; small expository refinements (clarifying unimodularity via lattices, and highlighting where avoidance enters uniqueness) would further enhance accessibility. Key steps and statements align with Theorem 1.7 and Sections 2, 4–6 of the paper (; ; ; ; ; ).