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On sets invariant under the action of the diagonal group

Elon Lindenstrauss, Barak Weiss

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Publication metadata

Journal citation: Ergodic Theory and Dynamical Systems 21(5), 1481–1500 (2001)
Publication date: 2001
DOI: 10.1017/S0143385701001717 ↗

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 · Author manuscript p. 2; proof pp. 16–18Classifies diagonal orbit closures containing a compact orbit by equal block groups and proves every strict enlargement is noncompact.
Theorem 3.2 · Author manuscript p. 5Moore ergodicity makes the diagonal action topologically transitive on the finite-volume algebraic support.

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Author-hosted preprint · explicit fallback for inaccessible Cambridge version of record

Elon Lindenstrauss and Barak Weiss. On sets invariant under the action of the diagonal group. Author-hosted preprint corresponding to Ergodic Theory and Dynamical Systems 21 (2001), no. 5, 1481–1500.

The Cambridge record identifies the exact version of record, but an openly retrievable publisher PDF was not obtained. The inspected fallback is the complete 20-page author-hosted preprint with the same title, authors, theorem numbering, and published pagination span; it is not represented as the Cambridge typeset object.

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Generated August 27, 2026

01Statements2 reported findingsNo statement issues found

Theorem 1.3 gives the equal-block classification and noncompactness conclusion used in EKL's Corollary 1.4.

Theorem 1.3No issue found

Orbit closures containing a compact diagonal orbit have equal-block homogeneous form

Author preprint p. 2 · Theorem 1.3; proof pp. 16–18 (corresponding published span pp. 1482 and 1496–1498)

For $G=SL(n,\mathbb R)$ , $\Gamma=SL(n,\mathbb Z)$ , and $F=\overline{Ay}$ containing a compact $A$ -orbit, the theorem writes $n=kd$ and identifies $F=Hy$ with $H$ a permutation-conjugate product of $d$ equal $GL(k,\mathbb R)$ blocks intersected with $G$ . It also states that if $F\ne Ay$ , then $F$ is noncompact. EKL uses exactly this classification after obtaining a compact $A$ -orbit in its algebraic support.

Theorem 3.2No issue found

Moore ergodicity supplies a dense diagonal orbit in the algebraic support

Author preprint p. 5 · Theorem 3.2; EKL06 VOR p. 545

The paper states and proves the needed general form of Moore's ergodicity theorem. In EKL's application it makes the $A$ -action topologically transitive on the finite-volume algebraic orbit, so that this orbit is an $A$ -orbit closure to which Theorem 1.3 applies.

02Proofs2 reported findingsFurther proof verification needed

The complete author proof is coherent through the represented Shah branch and terminal monographs, but Ratner's original 1991 proofs remain inaccessible.

Theorem 1.3 internal proofCorrect and complete modulo represented sources

The equal-block and noncompactness steps close internally modulo named sources

Author preprint pp. 5–11 and 16–18 · Theorems 3.2–3.4, Theorem 1.1, Lemma 6.2, Steps 6.1–6.3

Lemma 6.2 obtains the transitive Galois action attached to a compact $A$ -orbit. Step 6.1 rules out compact root-subgroup extensions; Step 6.2 combines Theorem 1.1, Theorem 3.4, and the Galois action to force equal block sizes; and Step 6.3 uses Mahler's compactness criterion to prove noncompactness whenever $H\ne A$ . The matrix, Galois, and compactness arguments are consistent.

Recursive source expansionClosed with represented article branch

Ratner 1991 and Shah 1991 are proof-critical non-book parents

Author preprint pp. 5–10 and 19–20 · Theorem 3.3, Theorem 3.4 and Appendix A

Theorem 1.1 uses Ratner's orbit-closure Theorems A–B through Theorem 3.3. Theorem 3.4 is derived in Appendix A from Shah 1991 Proposition 3.2 and Corollary 2.13, together with Ratner orbit closure and Margulis arithmeticity. Ratner and Shah are now represented as ordinary nodes; Shah's selected chain is recursively expanded, while Ratner's original 1991 proofs remain candidly unverified behind an access gate. The exact terminal book inputs are Raghunathan, Discrete Subgroups of Lie Groups, Lemma 1.14, Theorem 5.22, and Corollary 10.9; Zimmer, Ergodic Theory and Semisimple Groups, Theorem 2.2.6 and Chapter 6; Hoffman–Kunze, Linear Algebra, Theorem 6.12; and Onishchik–Vinberg, Lie Groups and Algebraic Groups, Section 6.3.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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