arXiv:2509.05272v1
Simultaneously bounded and dense orbits for commuting Cartan actions
math.DSmath.NT
Abstract
In this paper we prove that the set of points that have bounded orbits under one regular diagonal flow and dense orbits under the other diagonal flow commuting with the first one has full Hausdorff dimension in . To explain its application towards the Uniform Littlewood's Conjecture proposed in earlier work, we introduce the concept of ``fiberwise nondivergence'' for the action of a cone inside the full diagonal subgroup. Then our main result implies that there exists a dense subset of in which each point has a fiberwise non-divergent orbit under a cone inside the full diagonal subgroup and an unbounded orbit under every diagonal flow.
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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.
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Detailed mathematical audit
01Statements3 reported findingsCorrect
The full-dimension results for points with one bounded Cartan orbit and countably many dense or equidistributed commuting orbits, together with the fiberwise-nondivergence application, are correct; one ray parametrization has a harmless notation typo.
Theorems 1.1 and 1.2Correct
The mixed bounded-and-dense orbit statements are correct
Pages 3–4 and 8–16 · Theorems 1.1–1.2 · arXiv:2509.05272v1
Opposite Weyl chambers give complementary stable and unstable slices for Theorem 1.1. On the three-dimensional lattice space, high-entropy measures supported on bounded trajectories retain nonzero mass and positive entropy after averaging in any different Cartan direction; measure rigidity and slicing then give full Hausdorff dimension.
Full paper, version 1 ↗Theorem 1.7Correct
The fiberwise-nondivergence application is correct
Pages 5 and 16–18 · Theorem 1.7 · arXiv:2509.05272v1
A Baire-category construction finds points whose selected fibers repeatedly lie in one compact set, while a meager-set argument excludes bounded forward orbits for every Cartan ray. The condition that the compatible functional is not a root makes the kernel direction regular, as required by Theorem 1.2.
Ray parametrization in the proof of Lemma 4.2Typo · no status impact
The diagonal flow should be written with diag rather than an outer exp
Page 18 · proof that the bounded-ray set has empty interior · arXiv:2509.05272v1
The displayed ray applies exp to entries that are already exponentials. Replacing the outer exp notation by the diagonal matrix with those three entries gives the determinant-one flow used in the next line and in Mahler's criterion.
02Proofs2 reported findingsCorrect
The entropy-retention, measure-rigidity, slicing, and Baire-category arguments are correct and complete after the ray-notation typo is repaired.
Proof of Theorem 1.2Correct and complete
High entropy survives averaging and forces dense transverse orbits
Pages 10–16 · entropy argument for Theorem 1.2 · arXiv:2509.05272v1
The cusp-entropy bound prevents total escape of mass, positive leafwise entropy survives Cartan averaging, and the resulting positive-entropy Cartan components are Haar by measure rigidity. The dense-orbit conclusion holds simultaneously for the countable family, and Marstrand slicing restores the full ambient dimension.
Proof of Theorem 1.7Correct and complete after the notation correction
The nested-neighborhood and meagerness arguments close
Pages 16–18 · proof of Theorem 1.7 · arXiv:2509.05272v1
The neighborhoods can be chosen with nested compact closures, so their intersection is nonempty and avoids each prescribed nowhere dense set while retaining compact fibers at unbounded times. Rational points are dense and have unbounded orbits in every regular Weyl direction, proving that all bounded-ray sets form a meager union.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.