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 statement:
- Theorem 3.3 · VOR pp. 248–249 / PDF pp. 10–11; proof chain pp. 242–249 / PDF pp. 4–11Provides the finite invariant exhaustion argument reused in Dani 1984 Theorem 4.3.
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Exact reviewed source
Exact IAS repository Springer version-of-record scan · Inventiones Mathematicae 51 (1979), 239–260
S. G. Dani. On invariant measures, minimal sets and a lemma of Margulis. Inventiones Mathematicae 51 (1979), 239–260.
Open audited source ↗01Statements1 reported findingNo statement issues found
Theorem 3.3 gives exactly the invariant finite-measure exhaustion used in Dani 1984, Theorem 4.3, and hence in Shah's property-(D) input.
Theorem 3.3No issue found
A locally finite measure invariant under a unipotent subgroup is exhausted by finite invariant pieces
Version of record pp. 248–249 · PDF pp. 10–11 · Theorem 3.3
For a unipotent subgroup and a locally finite -invariant Borel measure on , the theorem produces countably many measurable -invariant sets of finite measure whose union is conull. The ergodic case is therefore finite. Dani 1984 explicitly reuses this induction to obtain the same property for general lattices, and Shah calls the resulting conclusion property (D).
02Proofs2 reported findingsNo proof issues found
The rank induction is complete in the exact article and has no further proof-critical ordinary-paper parent.
Theorem 3.3 proof chainNo issue found in the reviewed proof
Rank-one recurrence and the normal-series induction close internally
Version of record pp. 242–249 · PDF pp. 4–11 · Theorems 2.1, 2.10–2.11, 3.1–3.3
Theorem 2.1 proves a positive-density recurrence estimate by explicit polynomial and lattice arguments. Theorems 2.10–2.11 translate it to continuous and cyclic unipotent actions using Mahler compactness. Theorem 3.2 combines this recurrence with the individual ergodic theorem to produce an integrable positive invariant function in rank one. Theorem 3.3 then inducts along a normal series whose quotients are cyclic or one-parameter, averages the lower-rank invariant function, and turns it into a conull countable finite-measure exhaustion.
Terminal foundationsRecursively closed at terminal sources
The remaining imported inputs are standard or monographic
Version of record pp. 246–249 · PDF pp. 8–11
The exact named locator is Mahler's compactness criterion in Raghunathan, Discrete Subgroups of Lie Groups, Corollary 10.9. The individual ergodic theorem and the normal-series structure of unipotent groups are used as standard foundations and are stated in the article at the needed level. References to Margulis motivate the recurrence construction, but the polynomial recurrence proof is reproduced rather than imported. No additional ordinary journal-paper proof edge is required for the selected theorem.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.