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2501.17378

Dimension of diagonal self-affine measures with exponentially separated projections

Zhou Feng

correctmedium confidence
Category
Not specified
Journal tier
Strong Field
Processed
Sep 28, 2025, 12:56 AM

Audit review

The uploaded paper proves that for diagonal self-affine measures with simple Lyapunov spectrum and exponentially separated coordinate IFSs, one has dim μ = min{d, dimL(Φ,p)}, via a disintegration-based Ledrappier–Young formula for random measures and an entropy-increase argument that identifies an aggregate conditional entropy h_RW with H(p) in the exponentially separated setting (Theorems 1.12 and 1.3, and their reduction) . The candidate solution instead asserts a stronger, step-by-step saturation of the partial projection-entropies (h_j − h_{j−1} = min{χ_j, H − h_{j−1}}), attributing this directly to the paper; this step is neither stated nor proved in the paper, which works with a single “random-walk” entropy h_RW rather than per-level saturations. Hence the paper’s argument is correct and complete, while the model’s proof relies on an unsubstantiated claim.

Referee report (LaTeX)

\textbf{Recommendation:} no revision

\textbf{Journal Tier:} strong field

\textbf{Justification:}

The paper settles a conjecture in the diagonal self-affine setting under natural exponential separation by developing a Ledrappier–Young formula for random disintegrations together with an entropy-increase mechanism. The approach is robust and likely to influence subsequent work on non-conformal measures. The exposition is clear given the technical depth; the reduction to the main theorem is concise and self-contained.