arXiv:math/0210367v2
Extremal subspaces and their submanifolds
math.NTmath.DS11J83
Abstract
It is known that the properties of almost all points of R^n being not very well (multiplicatively) approximable are inherited by nondegenerate in R^n (read: not contained in a proper affine subspace) smooth submanifolds. In this paper we consider submanifolds which are contained in proper affine subspaces, and prove that the aforementioned diophantine properties pass from a subspace to its nondegenerate submanifold. The proofs are based on a correspondence between multidimensional diophantine approximation and dynamics of lattices in Euclidean spaces.
AI-generated audit
Audit summary
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
01Statements2 reported findingsCorrect
The extremality and strong-extremality inheritance theorems for nondegenerate submanifolds of affine subspaces, together with the explicit special-case coefficient criteria, are correct.
Theorems 1.2 and 1.4Correct
Extremality properties are inherited exactly as stated
Pages 3--5 · Theorems 1.2 and 1.4 · arXiv:math/0210367v2
For a good nondegenerate parametrization of an affine subspace, the exterior-algebra lower bounds governing quantitative nondivergence are the same as those for the affine parametrization itself. Thus an extremal, respectively strongly extremal, affine subspace transfers its property to almost every point of every nondegenerate submanifold. Conversely, failure of the relevant lower bound produces a uniform very-well-approximable, respectively multiplicatively approximable, obstruction at every point of the affine subspace.
Theorem 1.3Correct
The coefficient criterion is valid in both announced special cases
Page 4 and Section 4 (pages 14--18) · affine hyperplanes and lines through the origin · arXiv:math/0210367v2
For affine hyperplanes the only material exterior-power obstruction reduces to the ordinary Diophantine exponent of the coefficient matrix. For a line through the origin, transference identifies the same obstruction with the row vector's approximation class. In both cases the map is extremal exactly when , as stated.
02Proofs2 reported findingsCorrect
The quantitative-nondivergence criterion, its affine-subspace specialization, and the multiplicative exterior-power argument are correct and complete.
Sections 3--4Correct and complete
The exterior-power criterion is applied uniformly
Pages 9--18 · Theorems 3.8 and 4.3 and proof of Theorem 1.2 · arXiv:math/0210367v2
Goodness controls the small-value sets of all primitive exterior vectors, while nonplanarity supplies a uniform lower bound on a smaller ball. The quantitative nondivergence estimate is summable at every exponent strictly beyond Dirichlet's exponent. The affine parametrization calculation includes every exterior degree, so the inherited criterion is both necessary and sufficient in the stated class.
Section 5Correct and complete
The multiplicative reduction handles all coordinate subsets
Pages 19--25 · Theorems 5.3 and 5.5 · arXiv:math/0210367v2
The multiparameter diagonal action encodes the product inequalities, and the proof checks the lower bound for each primitive exterior vector before applying Borel--Cantelli. Zero coordinates are handled by the positive-part convention, so no missing coordinate case or boundary exponent remains.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.