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 statements:
- Section 3 conditional-measure construction · Annals pp. 173–179Leafwise conditional measures are coherently defined and transformed.
- Proposition 4.3 · Annals pp. 180–183Almost-sure Haar leafwise conditionals characterize global invariance.
- Appendix Theorem A.1 · Annals appendixThe maximal ergodic estimate used in EKL's measure construction holds with the stated constants.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
Annals of Mathematics version of record
Elon Lindenstrauss. Invariant measures and arithmetic quantum unique ergodicity. Annals of Mathematics 163 (2006), no. 1, 165–219.
Open audited source ↗01Statements2 reported findingsNo statement issues found
The selected conditional-measure invariance criterion and maximal ergodic estimate are correct.
Section 3 and Proposition 4.3No issue found
Leafwise Haar conditionals characterize global invariance
VOR pp. 173–183 · Theorem 3.6 and Proposition 4.3
The leafwise conditional measures are defined coherently through the flower atlas. Proposition 4.3 proves that almost-sure Haar conditionals are equivalent to global invariance under the acting locally compact group, as required by EKL.
Appendix Theorem A.1No issue found
The non-measure-preserving maximal inequality has the claimed weak type bound
VOR pp. 213–217 · Theorem A.1
For an expanding leaf action on a Besicovitch metric space, the theorem bounds the maximal function's level set by . EKL uses precisely this recurrence-control input.
02Proofs2 reported findingsNo proof issues found
The local-to-global invariance proof and the Besicovitch covering proof are complete.
Proof of Proposition 4.3No issue found in the reviewed proof
Local flower disintegrations patch to global group invariance
VOR pp. 183–185
Right-Haar leaf conditionals make integrals invariant for functions supported in a small flower center. A compact partition of unity extends this identity to every bounded compactly supported measurable function. The converse follows from uniqueness of disintegration.
Proof of Theorem A.1No issue found in the reviewed proof
Bounded-overlap covering gives the weak type estimate
VOR pp. 214–217
Lemma A.2 provides sufficiently large leafwise atoms by expanding a finite flower cover. Besicovitch selects balls covering the maximal set with uniformly bounded overlap. Integrating the selected-ball inequalities over the quotient atoms yields the printed constant and level-set bound.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.