Published paper
Role in dependence graphs
Proof-critical source
Winning property of counterexamples to Uniform Littlewood's Conjecture
This paper is included only for the following marked statement:
- Recurrence theorem and tower inequality · PNAS VOR pp. 651–655, equations (1)–(9)Supplies the first-return tower integral argument reused in Birkhoff's following ergodic-theorem paper.
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Exact reviewed source
Exact PNAS version of record · official PMC page scans · Proceedings of the National Academy of Sciences 17 (1931), 650–655
George D. Birkhoff. Proof of a recurrence theorem for strongly transitive systems. Proceedings of the National Academy of Sciences of the United States of America 17 (1931), no. 12, 650–655.
Open audited source ↗01Statements1 reported findingNo statement issues found
The recurrence note proves the almost-everywhere existence of a constant mean crossing time under its strong-transitivity hypothesis.
Recurrence theoremNo issue found
Successive crossings have an almost-everywhere asymptotic mean time
Version of record pp. 651–655 · theorem equation (1), proof equations (2)–(9)
For an analytic invariant-volume flow on a closed analytic manifold, a transverse analytic surface, and the printed strong-transitivity assumption, the th return time divided by converges almost everywhere to the constant ratio of total volume to flux through the section. This is exactly the tower estimate reused in Birkhoff's following ergodic-theorem paper.
02Proofs2 reported findingsNo proof issues found
The return-map and disjoint-tower proof is complete at standard recurrence and measure-theory foundations.
ProofNo issue found in the reviewed proof
Disjoint orbit towers rule out persistent deviations from the mean crossing time
Version of record pp. 652–655 · equations (2)–(9)
The first-return map preserves the transverse flux measure, so the integral of the th return time is exactly times total volume. Points with infinitely many upper deviations are decomposed into disjoint return towers; integrating over those towers would force an impossible strict excess over the known mean. The symmetric lower-deviation argument then gives almost-everywhere convergence.
Terminal foundationsClosed at published-book terminals
Poincaré recurrence and first-return facts terminate at the cited monograph
Version of record pp. 651–652 and footnote 1 · H. Poincaré, Méthodes nouvelles de la mécanique céleste, vol. 3
The note invokes Poincaré recurrence to assert that the first-return transformation is defined and one-to-one outside a null set. It cites Poincaré's monograph directly, and the rest of the proof is supplied in full. The unpublished von Neumann and Hopf work is contextual comparison, not a proof input, so no further ordinary-paper branch is opened.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.