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:
- Theorem 2.5 · Duke pp. 112–114Weighted bad approximation is equivalent to boundedness of the associated diagonal orbit.
- Corollary 3.5 · Duke pp. 118–119The matrix-flow result specializes to the weighted two-coordinate correspondence used by the focal paper.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
Version of record · Duke Mathematical Journal 95 (1998), no. 1, 107–124
Dmitry Y. Kleinbock. Flows on homogeneous spaces and Diophantine properties of matrices. Duke Mathematical Journal 95 (1998), no. 1, 107–124.
Open audited source ↗01Statements1 reported findingNo statement issues found
The selected weighted Dani correspondence is correct and has the exact quantifiers needed by Wu–Yang.
Theorem 2.5 and Corollary 3.5No issue found
Weighted bad approximation is equivalent to a bounded diagonal orbit
Duke article pp. 112–114 and 118–119 · Theorem 2.5 and Corollary 3.5
Specializing the weight vectors to and the one-column matrix identifies the printed loose/badly-approximable condition with and the diagonal semigroup with . Corollary 3.5 then gives boundedness of exactly that orbit.
02Proofs2 reported findingsNo proof issues found
The balancing argument and Mahler compactness criterion prove both directions.
Proof of Theorem 2.5No issue found in the reviewed proof
The time parameter balances the weighted coordinate inequalities
Duke article pp. 112–114
An integer approximation vector is pushed by the diagonal flow so that its expanding and contracting components balance at the chosen time. Conversely, any short vector along the orbit yields an integer solution violating the weighted lower bound. The exponents and determinant-one normalization agree.
Corollary 3.5No issue found in the reviewed proof
Mahler compactness closes the orbit-boundedness equivalence
Duke article pp. 118–119
Theorem 2.5 turns the Diophantine lower bound into a uniform lower bound for nonzero lattice vectors. Raghunathan Corollary 10.9, Mahler's compactness criterion, is then exactly the compactness input needed.
03Novelty0 reported findingsNo novelty concerns found
No non-novelty findings.