2110.14466
Random Lochs’ Theorem
Charlene Kalle, Evgeny Verbitskiy, Benthen Zeegers
correctmedium confidence
- Category
- math.DS
- Journal tier
- Specialist/Solid
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper’s Random Lochs’ Theorem (Theorem 1.1) proves lim mT,S(n,ω,ω̃,x)/n = hfib(T)/hfib(S) by combining a fiber SMB-type cylinder-size law (Theorem 1.3(i)) with Dajani–Fieldsteel’s general partition theorem (Theorem 1.2). The candidate solution reproduces the same strategy at a proof-idea level: establish exponential cylinder-size asymptotics and then use standard interval-partition geometry plus a Borel–Cantelli boundary argument to sandwich m between two affine functions of n. This mirrors the classical proof underlying Theorem 1.2, and the paper even uses the same boundary-neighborhood truncation idea elsewhere (cf. Lemma 3.2’s construction of C). No substantive logical gap or contradiction was found.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions
\textbf{Journal Tier:} specialist/solid
\textbf{Justification:}
The result is established cleanly by pairing a fiber SMB cylinder-size law with a general partition comparison theorem. The assumptions and constructions (conditional measures, generating partitions, finiteness conditions) are handled carefully. A few small clarifications linking assumptions to specific lemmas would improve readability, but the substance is correct and well-motivated.