Published paper
Abstract
The paper develops a characterization of Haar measure and an invariant-measure classification framework for maximal horospherical flows on noncompact homogeneous spaces.
Role in dependence graphs
Proof-critical source
The Dirichlet spectrum
This paper is included only for the following marked statement:
- Theorem 2.4 and Proposition 9.3 · Inventiones Mathematicae 47 (1978), selected classification branchHaar-measure characterization and the invariant-measure step used by Dani 1981 Theorem 9.1.
Proof-critical source
Winning property of counterexamples to Uniform Littlewood's Conjecture
This paper is included only for the following marked statement:
- Lemma 8.2 · Printed p. 130 / PDF p. 30Density of in makes the action ergodic on ; Shah adapts the same translation-continuity argument.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
Exact IAS repository Springer version-of-record scan · Inventiones Mathematicae 47 (1978), 101–138
S. G. Dani. Invariant measures of horospherical flows on noncompact homogeneous spaces. Inventiones Mathematicae 47 (1978), no. 2, 101–138.
Open audited source ↗01Statements4 reported findingsNo statement issues found
The selected Theorem 2.4 and Proposition 9.3 are correct. Theorem 2.4 needs the uniquely determined union-to-intersection repair to the definition of its set . Theorem 10.1's conclusion is true and was later reproved, but the proof printed here is invalid in the stated product generality.
Selected Theorem 2.4Correct after a uniquely determined notation repair
The Haar-measure characterization is correct after the published notation repair
Statement on printed page 109 · PDF page 9; correction identified in 1981 Remark 3.6, printed page 366 · 1981 PDF page 10
The 1978 setup prints as the union of the cusp cells . The proof requires their intersection, and the author's 1981 Remark 3.6 explicitly makes that replacement. With the corrected , the theorem states and proves that a lattice-invariant, lattice-finite ergodic measure supported on this full-cusp region is -invariant.
1981 correction notice, Remark 3.6 ↗Selected Proposition 9.3Correct · focal dependency verified
Homogeneity lifts correctly from the semisimple quotient
Printed pages 132–134 · PDF pages 32–34
Properness of the quotient projection first confines the measure to the preimage of the homogeneous orbit. The proof then constructs a subgroup containing the horospherical group, disintegrates along the compact kernel fibers, and smooths those conditionals by the kernel action. Absolute continuity with respect to the homogeneous measure, followed by ergodicity, yields full subgroup invariance and the asserted closed finite-volume orbit.
Theorem 10.1 and Theorem AConclusion appears correct · suspected proof issue in full generality
The conclusions are retained, but not by the printed general proof
Theorem A on printed page 102 · PDF page 2; Theorem 10.1 and proof on printed pages 134–137 · PDF pages 34–37; 1981 Remark 8.3 on printed page 381 · 1981 PDF page 25
The author's 1981 Remark 8.3 says that the proof of Theorem 10.1 is valid for lattices in real-rank-one groups but not in the broader product scope stated here because Lemma 10.3 is incorrect. The 1981 Theorem 8.2 supplies a new proof covering the general semisimple case, so the classification conclusion is not false; it is simply not established by the 1978 argument in that generality.
1981 Remark 8.3 and replacement Theorem 8.2 ↗Lemma 8.2 · Shah dependencyNo issue found
Dense normal-subgroup action is ergodic on a finite-volume quotient
Printed page 130 · PDF page 30 · Lemma 8.2; adapted in Shah 1991 Lemma 2.10 on printed page 319
If carries a finite -invariant measure, is normal in , and is dense in , Lemma 8.2 proves that the -action is ergodic. Shah uses the same right-translation continuity argument for a locally finite semi-invariant measure, replacing the function space by locally integrable functions. The selected statement is independent of the later Section 10 defect.
02Proofs5 reported findingsSuspected incomplete or incorrect proofs
The selected Theorem 2.4 and Proposition 9.3 have valid proof chains and do not depend on the later faulty reduction. The proof of Theorem 10.1 is nevertheless invalid in its stated product generality because Lemma 10.3 is false, exactly as the author recorded in 1981.
Proof of selected Theorem 2.4Correct and complete after the notation repair
The boundary and reduction-theory argument closes after correcting
Printed pages 109–128 · PDF pages 9–28 · Sections 3–7
Sections 3 and 4 reduce -invariance to an absolute-continuity criterion for the boundary measure. Section 5 turns the lattice fundamental-domain decomposition into the required square-integrability estimate, while Sections 6 and 7 establish the convolution comparison and Proposition 5.14. These steps use the intersection version of later identified in Remark 3.6 and do not use Lemma 10.3.
Proof of selected Proposition 9.3No issue found in the reviewed proof
Disintegration over the proper quotient supplies the lifted invariant measure
Printed pages 132–134 · PDF pages 32–34
Lemmas 9.1 and 9.2 provide the proper projection and the needed ergodicity after passing to the normal semisimple subgroup. The constructed subgroup maps onto the homogeneous subgroup downstairs. Convolution by compactly supported kernel functions makes the conditional measures absolutely continuous with respect to Haar measure on the lifted orbit, and ergodicity forces equality up to scale. This branch is logically prior to and independent of Section 10.
Lemma 10.3 and proof of Theorem 10.1Suspected incorrect and incomplete as printed
The reduction to a simple rank-one noncompact block is false
Lemma 10.3 on printed pages 134–135 · PDF pages 34–35; proof of Theorem 10.1 on printed pages 136–137 · PDF pages 36–37; correction in 1981 Remark 8.3
Lemma 10.3 claims that every noncompact irreducible lattice block in a product of rank-one factors must itself have only one simple factor. That reduction is false, and the ensuing induction repeatedly uses it to split the lattice and isolate a simple rank-one factor. The author's 1981 Remark 8.3 explicitly identifies Lemma 10.3 as incorrect and states that the proof of Theorem 10.1 is invalid in its full stated generality. The later Theorem 8.2 replaces, rather than validates, this printed step.
1981 Remark 8.3 ↗Dependency separationSelected results unaffected
The selected graph edges remain valid despite the Section 10 defect
Theorem 2.4 on printed pages 109–128 and Proposition 9.3 on printed pages 132–134, both preceding Section 10
Theorem 2.4 is completed in Sections 3–7, and Proposition 9.3 is completed within Section 9. Neither invokes Lemma 10.3 or Theorem 10.1. Their use in the 1981 proof of Theorem 9.1 is therefore not contaminated by the later invalid reduction.
Proof of Lemma 8.2 · Shah dependencyNo issue found in the reviewed proof
The continuity argument closes directly and survives Shah's stated modification
Printed page 130 · PDF page 30; Shah 1991 printed page 319 · Lemma 2.10
A -invariant function on lifts to a locally integrable function on that is invariant under right translation by both and . Continuity of right translation in the local-integrability topology and density of then make it -invariant and hence constant. This is precisely the feature Shah retains for a locally finite semi-invariant measure. No further ordinary-paper proof parent is used in this selected branch.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.