Published paper
Abstract
For rank-one homogeneous spaces, the paper proves a quantitative compact-return alternative for unipotent trajectories using reduction theory and an exterior-power representation.
Role in dependence graphs
Proof-critical source
The Dirichlet spectrum
This paper is included only for the following marked statement:
- Lemma 2.4 · ETDS 4 (1984), pp. 28–31Rank-one compact-return alternative used in Shah 1996 Theorem 2.2.
Proof-critical source
Winning property of counterexamples to Uniform Littlewood's Conjecture
This paper is included only for the following marked statement:
- Theorem 4.3 · Printed p. 33 / PDF p. 9; proof final paragraph; exact 1986 correction checkedEvery locally finite measure invariant under a unipotent subgroup is exhausted by finite invariant pieces, Shah's property (D).
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
Exact Cambridge versions of record · article and published correction
S. G. Dani. On orbits of unipotent flows on homogeneous spaces. Ergodic Theory and Dynamical Systems 4 (1984), no. 1, 25–34; correction, Ergodic Theory and Dynamical Systems 6 (1986), 321.
Open audited source ↗01Statements5 reported findingsSuspected incorrect statements
The selected Lemma 2.4 and the main recurrence conclusions are correct after local proof repairs. Lemma 2.5 is wrong as printed but corrected in 1986, and Lemma 2.7 has self-contradictory endpoint hypotheses with a unique mechanical repair.
Selected Lemma 2.4Correct · focal dependency verified
One lattice representative works throughout each interval
Statement on printed page 29 · PDF page 5 · exact 1984 Cambridge version of record
For an interval on which the orbit remains in the fixed cusp neighborhood, Corollary 2.3 makes the admissible representation vector locally unique up to sign. Connected continuation therefore supplies a single lattice element throughout the interval. The 1986 correction changes neither this statement nor its input.
1984 Cambridge version of record ↗Lemma 2.5Suspected incorrect as printed · corrected post-publication
The boundedness equivalence has the wrong fixed vector
Printed page 30 · PDF page 6; corrected on ETDS 6 (1986), page 321 · correction PDF page 1
The 1984 statement says boundedness of is equivalent to . Constancy of the coordinate polynomials gives only . Item (1) of the exact 1986 correction makes precisely that replacement and confirms that the printed proof already proves the corrected form.
1986 Cambridge correction ↗Lemma 2.7Suspected incorrect as written · an apparently sound repair
The endpoint hypotheses are self-contradictory
Printed page 31 · PDF page 7; application on printed pages 31–32 · PDF pages 7–8
The lemma assumes both and for every , so the displayed class is empty. Replacing the strict inequality by on , or replacing that interval by , makes the compactness proof valid. The application has equality at the initial endpoint and strict inequality thereafter, so this unique endpoint repair changes no conclusion.
Theorem 0.2 and Proposition 1.2Correct after apparently sound repairs
The compact-return conclusions survive all local repairs
Printed pages 26–33 · PDF pages 2–9
After the repairs to Lemmas 2.4, 2.5, and 2.7, the cusp decomposition, exterior-power height, bounded-degree polynomial root count, and interval-length estimate give the claimed compact-return frequency. Summing over the finite set of cusps yields Theorem 0.2 with the stated uniform error.
Theorem 4.3 · Shah dependencyCorrect after apparently sound ancestor repairs
Every locally finite measure invariant under a unipotent subgroup has a countable finite-measure exhaustion
Statement on printed page 33 · PDF page 9; cited in Shah 1991 Proposition 2.5 on printed page 318
For a subgroup consisting of unipotent elements, Theorem 4.3 states that every locally finite -invariant measure on admits a countable exhaustion by measurable -invariant sets of finite measure; in particular every locally finite ergodic -invariant measure is finite. This is exactly Shah's property (D). The conclusion survives the exact 1986 repair to Theorem 4.1, and the omitted induction is verified in the represented Dani 1979 Theorem 3.3 source.
02Proofs5 reported findingsSuspected incomplete or incorrect proofs
The selected Lemma 2.4 has a false boundary assertion in its printed proof but admits a short connected-continuation repair. The 1986 correction repairs Corollary 2.6 and identifies the invalid irreducible-lattice reduction in Theorem 4.1; Corollary 2.3 and Lemma 2.7 also need local formal repairs.
Proof of selected Lemma 2.4Correct after an apparently sound local repair
The proof assumes an unstated boundary condition on an arbitrary interval
Statement on printed page 29 · PDF page 5; proof on printed page 30 · PDF page 6
For an arbitrary interval , the printed assertion that belongs to exactly when is false because need not be an endpoint of a maximal visit to . Choose an interior time and one witnessing lattice element. The times for which its vector has norm below form a nonempty open set. Corollary 2.3 identifies that fixed vector, up to sign, with every local witness whose norm is below ; continuity makes the set closed. Connectedness gives the whole interval, and continuity also gives the closed-interval version. This proves the stated fixed-representative conclusion without a boundary assumption.
1984 Cambridge version of record ↗Corollary 2.3Suspected minor formal issue
The character is applied outside its stated domain
Printed page 29 · PDF page 5
The proof writes the character on each lattice element although the character was defined only on the conjugated parabolic and neither element is shown to lie there. For their quotient , however, the preceding separation result gives and the representation preserves the line spanned by . Hence its real scalar satisfies , so the two vectors agree up to sign. This directly restores the claimed well-definedness.
Corollary 2.6Suspected incorrect as printed · corrected post-publication
The orbit containment must use the corrected Lemma 2.5
Printed page 30 · PDF page 6; correction page 321 · correction PDF page 1, item (1)
The 1984 proof incorrectly derives . The corrected lemma instead places in the compact-cusp coset based at for every . Theorem 1.1(ii) makes its quotient compact, so the same contradiction with an unbounded orbit proves Corollary 2.6. The published correction states this repaired containment explicitly.
1986 Cambridge correction, item (1) ↗Theorem 4.1Suspected incomplete as printed · published repair supplied
The reduction to an irreducible lattice is invalid
Printed page 33 · PDF page 9; correction page 321 · correction PDF page 1, item (2)
Separate positive-return conclusions in the factors do not by themselves give simultaneous positive return in their product, so the printed reduction without loss to an irreducible lattice is invalid. Item (2) of the 1986 correction acknowledges the error and explains that the full one-minus-error estimate of Theorem 0.2 repairs the product step; it also records an alternative derivation from Dani's 1986 sequel. The theorem is retained, but the 1984 proof is not complete on its own.
1986 Cambridge correction, item (2) ↗Proof of Theorem 4.3 · Shah dependencyCorrect after apparently sound ancestor repairs
The cited property-(D) induction is verified in Dani 1979
Printed page 33 · PDF page 9 · final paragraph of Theorem 4.3
The article says that Theorem 4.3 follows from Theorem 4.1 and the individual ergodic theorem by following almost verbatim the proof of Dani 1979, Theorem 3.3, and omits the details. Theorem 4.1 is read with the exact 1986 repair already recorded above. Dani 1979 Theorem 3.3 and its proof on pp. 248–249 are now represented and verified, closing the selected property-(D) induction. The paper-wide adverse proof status remains warranted by the unrelated printed defects documented in the other findings.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.