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:
- Theorem 1.3 · Annals pp. 515–516; proof pp. 532–548Positive entropy for the diagonal action forces the algebraic invariance needed in the Littlewood-exception argument.
- Corollary 1.4 · Annals pp. 516 and 544–545A positive-entropy invariant measure cannot be compactly supported.
- Theorem 1.5 · Annals pp. 516–517 and 558The classical Littlewood exceptional set has Hausdorff dimension zero.
- Theorem 10.1 · Annals pp. 548–555Positive transversal dimension produces an invariant probability measure with positive entropy.
- Proposition 11.1 · Annals pp. 557–558Littlewood's inequality is equivalent to boundedness of the positive diagonal orbit.
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Exact reviewed source
Annals of Mathematics version of record
Manfred Einsiedler, Anatole Katok, and Elon Lindenstrauss. Invariant measures and the set of exceptions to Littlewood's conjecture. Annals of Mathematics 164 (2006), no. 2, 513–560.
Open audited source ↗01Statements2 reported findingsNo statement issues found
The Littlewood-exception dimension theorem and the lattice-orbit correspondence selected by Wu–Yang are correct.
Theorem 1.5No issue found
The Littlewood exceptional set has Hausdorff dimension zero
VOR pp. 516–517 and 558 · Theorem 1.5 and its proof
The exceptional pairs embed into the bounded -orbit set in by Proposition 11.1. The transversal dimension-zero theorem then gives the claimed Hausdorff dimension zero. This is the exact result used by Wu–Yang.
Proposition 11.1No issue found
Littlewood's inequality is equivalent to boundedness of the positive diagonal orbit
VOR pp. 557–558 · Proposition 11.1
The proposition expands a short-vector condition for and balances the two diagonal parameters against the integer denominator. Both implications reproduce the product with the required uniform lower bound.
02Proofs2 reported findingsFurther proof verification needed
The internal dimension and correspondence arguments are correct. The recursive classification and compact-support contradiction are explicit, but the original proofs of three material ancestors—Margulis–Tomanov 1996, Prasad–Raghunathan 1972, and Ratner 1991—could not be checked in proof-bearing copies.
Theorem 1.5 proof chainCorrect and complete modulo represented sources
The entropy-to-dimension argument closes through Theorem 10.1, Theorem 1.3, and Corollary 1.4
VOR pp. 544–558 · Corollary 1.4 and Sections 10–11
A positive-dimensional compact transversal would support an invariant probability measure with positive entropy by the variational construction in Theorem 10.1. Theorem 1.3 makes that measure algebraic. The needed contradiction is then precisely Corollary 1.4—not algebraicity by itself: Prasad–Raghunathan supplies a compact Cartan orbit inside the algebraic support, and Lindenstrauss–Weiss classifies the resulting diagonal orbit closure and makes it noncompact when the positive-entropy support is larger than that Cartan orbit. Proposition 11.1 transfers the dimension result to Littlewood exceptions. The entropy estimates, compact-support contradiction, and covering steps align.
Recursive source-access limitsNot independently verified three imported proofs
Three exact original proofs remain inaccessible
VOR pp. 532–545 · Theorem 1.3 and Corollary 1.4 source chains
EKL restates the needed Margulis–Tomanov Theorems (a) and (b) and checks their hypotheses, but the exact 1996 target article remained behind HTML gates. EKL also gives the exact role of Prasad–Raghunathan Theorem 2.13, while the exact 1972 Annals proof remained gated. Finally, the accessible Lindenstrauss–Weiss proof closes through Ratner's 1991 orbit-closure theorems; the exact Ratner target was gated, so only a truthfully disclosed related-publication witness is available. No contradiction or hypothesis mismatch was found, and all other selected ancestors are represented recursively.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.