arXiv:1912.13070v2
Singular vectors on manifolds and fractals
math.NTmath.DS11J1311J5437A17
Abstract
We generalize Khintchine's method of constructing totally irrational singular vectors and linear forms. The main result of the paper shows existence of totally irrational vectors and linear forms with large uniform Diophantine exponents on certain subsets of , in particular on any analytic submanifold of of dimension which is not contained in a proper rational affine subspace.
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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 abstract construction of totally irrational uniformly approximable vectors and its manifold, fractal, weighted, and exponent consequences are correct; one weighted transference display has a harmless repeated-star typo.
Theorem 1.1Correct
The abstract approximation theorem is correct
Pages 4–6 and 9–13 · Theorem 1.1 and proof · arXiv:1912.13070v2
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 guarantees that the approximation scales tend to infinity, and branching gives uncountably many points.
Full paper, version 2 ↗Theorems 1.6 and 1.7Correct
The fractal and analytic-manifold applications satisfy the abstract hypotheses
Pages 7–8 and 13–20 · Theorems 1.6–1.7 · arXiv:1912.13070v2
Dense rational coordinates verify the product-set incidence conditions. For analytic manifolds, the stratification of rational-hyperplane sections and the reduction to a non-rational analytic surface supply the required connected components and avoidance property.
Weighted transference displayTypo · no status impact
The conclusion should use the unstarred weighted exponent
Page 6 · display preceding Corollary 1.5 · arXiv:1912.13070v2
The premise and conclusion in the displayed implication both carry a star, but the following corollary and the cited transference result use an unstarred exponent in the conclusion. Removing the star from that conclusion is the unique reading consistent with the deduction.
02Proofs2 reported findingsCorrect
The nested approximation construction and the verification for products and analytic manifolds are correct and complete after the notation correction.
Proof of Theorem 1.1Correct and complete
Approximation and total irrationality are enforced simultaneously
Pages 9–13 · proof of Theorem 1.1 · arXiv:1912.13070v2
At each step a new resonant component meeting the current neighborhood is chosen outside the next forbidden set. Continuity gives the desired approximation throughout a smaller neighborhood, and the enumeration of rational hyperplanes ensures that every limit point is totally irrational.
Proofs of Theorems 1.6 and 1.7Correct and complete after the notation correction
The geometric incidence conditions are fully checked
Pages 13–20 · product and manifold proofs · arXiv:1912.13070v2
Perfectness prevents isolated branches in the product case. In the analytic case, Baire category reduces dimension without introducing rational containment, and the semianalytic section decomposition verifies coverage, density, and finite-avoidance for every rational hyperplane.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.