Published paper
Abstract
The paper constructs totally irrational singular vectors and linear forms with large uniform Diophantine exponents on manifolds, fractals, and other prescribed sets.
Role in dependence graphs
Proof-critical source
On Some Properties of Irrational Subspaces
This paper is included only for the following marked statement:
- Proposition 5.1 · arXiv:1912.13070v2, §5Bounds the ordinary exponent of vectors lying in a badly approximable subspace and combines it with the uniform/ordinary ratio estimate.
AI-generated audit
Audit summary
Not a correctness certificate. These reports do not replace expert scrutiny or formal verification.
Exact reviewed source
arXiv:1912.13070v2 · explicit fallback for inaccessible version of record
Dmitry Kleinbock, Nikolay Moshchevitin, Barak Weiss. Singular vectors on manifolds and fractals. arXiv:1912.13070v2.
The Israel Journal of Mathematics version of record could not be retrieved from the publisher in this audit session; the exact arXiv v2 manuscript was reviewed and is not represented as the version of record.
Open audited source ↗01Statements4 reported findingsCorrect
The exact arXiv v2 fallback supports the abstract construction of totally irrational singular vectors, its product and analytic-manifold applications, and the upper bounds for uniform exponents on badly approximable affine subspaces. Two reversed index pairs and one repeated star are uniquely repairable notation errors.
Theorem 1.1 and Corollaries 1.2-1.5Correct
The abstract construction supplies the claimed singular and weighted consequences
arXiv:1912.13070v2, pp. 4-6 and 9-13, Theorem 1.1, Corollaries 1.2-1.5, and proof
The four incidence and density hypotheses permit a nested choice that alternately realizes arbitrarily strong rational relations and avoids every rational affine hyperplane. Properness of the height function makes the approximation scales tend to infinity, and the branching hypothesis gives uncountably many limit points. The stated dual, weighted, and exponent corollaries then follow from the explicitly cited transference implications.
Theorems 1.6 and 1.7Correct
The product and analytic-manifold applications meet the abstract hypotheses
arXiv:1912.13070v2, pp. 7-8 and 13-20, Theorems 1.6-1.7 and proofs
Dense rational coordinates verify the product-set incidence conditions. For analytic manifolds, the reduction to a non-rational analytic surface and the decomposition of rational-hyperplane sections into basic components supply the required coverage, density, and finite-avoidance properties. The correction to an earlier connected-component argument is incorporated into the proof given here.
Proposition 5.1Correct
Badly approximable affine subspaces impose both announced uniform-exponent bounds
arXiv:1912.13070v2, pp. 15-17, Proposition 5.1, Lemma 5.3, and proof
Writing the affine subspace through its augmented matrix converts bad approximability into the lower bound needed for . For a totally irrational point, the independently checked Marnat-Moshchevitin estimate , combined with that ordinary-exponent bound, gives the polynomial defining and the sharper conclusion. The Marnat-Moshchevitin theorem is the only external paper result materially required by this marked argument; the Nguyen-Poels-Roy reference is offered only as an alternative proof of the same input.
Weighted display and Lemma 5.3Typo · no status impact
Three subscripts or decorations are mechanically reversed
arXiv:1912.13070v2, p. 6, display preceding Corollary 1.5; pp. 16-17, Lemma 5.3, equation (5.7), and proof
The weighted implication must have the unstarred exponent in its conclusion, as Corollary 1.5 and the cited transference theorem require. In Lemma 5.3 and its proof, every printed must be : only is defined, and equations (5.5) and (5.9) already use that order. These local corrections are unique and leave the deductions unchanged.
02Proofs3 reported findingsCorrect
The nested construction, the geometric verification for products and manifolds, and the badly approximable-subspace argument are correct and complete after the three local notation repairs. The external ratio estimate needed for Proposition 5.1 was checked against the audited Marnat-Moshchevitin source.
Proof of Theorem 1.1Correct and complete
Approximation and total irrationality are enforced simultaneously
arXiv:1912.13070v2, pp. 9-13, proof of Theorem 1.1
At each stage a new resonant component meeting the current neighborhood is chosen outside the next forbidden set. Continuity preserves the desired approximation on a smaller neighborhood, while enumeration of rational hyperplanes forces every limit point to be totally irrational. The diameter and proper-height choices close the limiting argument.
Proofs of Theorems 1.6 and 1.7Correct and complete
The incidence and avoidance checks cover the claimed geometric settings
arXiv:1912.13070v2, pp. 13-15 and 17-20, product and analytic-manifold proofs
Perfectness prevents isolated branches in the product case. In the analytic case, Baire category reduces dimension without introducing rational containment, and the basic-component formulation avoids the flaw in the earlier connected-component proof while preserving all four abstract hypotheses.
Proof of Proposition 5.1Correct and complete after notation correction
The affine-matrix reduction and transference algebra close after the subscript repair
arXiv:1912.13070v2, pp. 16-17, proof of Proposition 5.1
The augmented integer vector has norm comparable to its denominator, so a hypothetical improvement contradicts bad approximability and proves part (i). In part (ii), the Marnat-Moshchevitin root equation is transformed correctly into ; monotonicity on the relevant interval then permits replacing by and yields exactly the defining inequality for .
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.