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arXiv:1501.05409v3

Bounded orbits of diagonalizable flows on SL3(R)/SL3(Z)\rm{SL}_3(\mathbb{R})/\rm{SL}_3(\mathbb{Z})

Jinpeng An, Lifan Guan, Dmitry Kleinbock

math.DS37A1711J13
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Paper metadata

Original submission
January 22, 2015
Latest revision
January 31, 2015 · version 3
Claimed MSC
37A17, 11J13
arXiv categories
math.DS
Comments
21 pages. Added references and expanded the last section

Abstract

We prove that for any countably many one-parameter diagonalizable subgroups FnF_n of SL3(R)\rm{SL}_3(\mathbb{R}), the set of Λ∈SL3(R)/SL3(Z)Λ\in\rm{SL}_3(\mathbb{R})/\rm{SL}_3(\mathbb{Z}) such that all the orbits FnΛF_nΛ are bounded has full Hausdorff dimension.

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Audit summary

Audited against arXiv v3

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

Current report

Detailed mathematical audit

Generated August 19, 2026
01Statements2 reported findingsCorrect

The hyperplane-absolute-winning and thickness theorems for bounded diagonalizable orbits on the three-dimensional lattice space are correct; the recalled game rule has a harmless radius typo.

Theorems 1.1–1.3Correct

The bounded-orbit sets have the stated winning and intersection properties

Pages 3–5 and 10–25 · Theorems 1.1–1.3 · arXiv:1501.05409v3

The unstable-horospherical slice is shown to be hyperplane absolute winning for every diagonalizable direction under consideration. Local product coordinates and invariance of the game then give thickness for countable intersections in the full homogeneous space.

Full paper, version 3 ↗
Recalled Schmidt-game ruleTypo · no status impact

Bob's center bound should use Alice's radius

Page 6 · definition of Schmidt's game · arXiv:1501.05409v3

Alice's ball has radius equal to alpha times Bob's preceding radius, but the printed center bound for Bob's next ball uses the preceding Bob radius again. Replacing it by Alice's radius is the unique standard rule that guarantees containment, and the subsequent proof uses that corrected nested-ball rule.

02Proofs2 reported findingsCorrect

The hyperplane-potential strategy and local-product reduction are correct and complete after the recalled-rule typo is corrected.

Theorem 3.1Correct and complete

The potential-game strategy controls every short-vector obstruction

Pages 10–19 · Theorem 3.1 and proof · arXiv:1501.05409v3

Dangerous lattice vectors are organized by height, vectors relevant to one scale satisfy a common linear constraint, and the total potential of the deleted hyperplane neighborhoods remains within the legal budget. The surviving trajectory stays in a compact set by Mahler's criterion.

Deduction of Theorems 1.1–1.2Correct and complete after the notation correction

The slice-to-space argument closes

Pages 19–25 · completion of the main results · arXiv:1501.05409v3

Winning on the expanding horospherical subgroup is preserved under the local coordinate maps, while stable and neutral factors contribute their full dimensions. Countable intersections retain the same winning parameter on each chart.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1501.05409v3
Authors listed
Jinpeng An, Lifan Guan, Dmitry Kleinbock
Audit date
August 19, 2026
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