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:
- Lemma 2.2 · VOR p. 317Produces the unique smallest subgroup with the prescribed closed orbit.
- Corollary 2.13 · VOR pp. 320–321A dense unipotent-generated orbit forces containment in the stabilizer's Zariski closure.
- Proposition 3.2 · VOR pp. 321–322The minimal rational algebraic closure has unipotent radical and recovers the connected orbit group.
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Version of record · Mathematische Annalen 289 (1991), 315–334
Nimish A. Shah. Uniformly distributed orbits of certain flows on homogeneous spaces. Mathematische Annalen 289 (1991), 315–334.
Open audited source ↗01Statements3 reported findingsNo statement issues found
Lemma 2.2, Corollary 2.13, and Proposition 3.2 give the subgroup-minimality, Zariski-closure, and arithmetic algebraicity conclusions used in Lindenstrauss–Weiss's proof of Theorem 3.4.
Lemma 2.2No issue found
A unique smallest closed subgroup has the prescribed closed orbit
Version of record p. 317 · Lemma 2.2
For two closed homogeneous orbits, every orbit of the intersection subgroup in their intersection is both open and closed. Consequently, for each point there is a unique smallest Lie subgroup containing the given subgroup for which is closed. Lindenstrauss–Weiss uses this minimal subgroup formulation when comparing lifted and projected orbit closures.
Corollary 2.13No issue found
Orbit-density forces containment in the Zariski closure of the stabilizer
Version of record pp. 320–321 · Proposition 2.11, Corollaries 2.12–2.13
When is the identity component of the real points of an algebraic matrix group, is connected, is generated by algebraic unipotent one-parameter subgroups, and is dense in , the paper proves that is contained in the Zariski closure of the stabilizer . This is assertion 2 in the Lindenstrauss–Weiss formulation of Theorem 3.4.
Proposition 3.2No issue found
The minimal rational algebraic closure has unipotent radical and recovers the orbit group
Version of record pp. 321–322 · Proposition 3.2
For an arithmetic lattice in the real identity component of a -group and a subgroup generated by algebraic unipotent one-parameter subgroups, the smallest algebraic -group containing the subgroup has unipotent radical and its real identity component is the connected group defining the closed orbit. Lindenstrauss–Weiss cites these as assertions 1 and 3 of Theorem 3.4.
02Proofs2 reported findingsFurther proof verification needed
The selected Shah arguments are internally coherent through represented parents; the warning remains because Moore's branch ends at inaccessible Dani 1977 and Moore 1966 proofs.
Internal selected chainCorrect and complete modulo represented sources
The subgroup and algebraic reductions close internally modulo named sources
Version of record pp. 317–322 · Lemma 2.2, Theorem 2.3, Proposition 2.11, Corollaries 2.12–2.13, Proposition 3.2
Lemma 2.2 is proved directly. Theorem 2.3 supplies finite volume and ergodicity for the minimal subgroup, Proposition 2.11 proves the required Borel-density variant by a projective representation argument, and Corollaries 2.12–2.13 then yield the stabilizer statement. Proposition 3.2 combines that result with the rational-algebraic and lattice facts from Raghunathan's monograph. The logic matches the claims imported by Lindenstrauss–Weiss.
Recursive source expansionClosed with represented article branch
Dani 1978, Dani 1984, and Moore 1980 are genuine non-book proof parents
Version of record pp. 318–320 · Propositions 2.5 and 2.7, Lemma 2.10, proof of Theorem 2.3
Shah's Proposition 2.5 imports Dani 1984, Theorem 4.3, for property (D); Proposition 2.7 imports Moore 1980, Theorem 1.1, for the Mautner phenomenon; and Lemma 2.10 imports the proof of Dani 1978, Lemma 8.2, with a stated modification. All three are represented. Dani 1978 closes for the selected lemma, and Dani 1984's property-(D) chain closes through exact Dani 1979 Theorem 3.3 despite unrelated adverse findings elsewhere in Dani 1984. Moore 1980 is represented, but its decisive Dani 1977 and Moore 1966 proofs remain inaccessible and explicitly unverified. The cited Margulis remark is followed by a proof, the Dani–Margulis citation in Proposition 2.11 is only comparative, and the remaining direct algebraic and lattice inputs terminate at exact Raghunathan monograph locators.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.