arXiv:1509.05439v3
Intrinsic Diophantine approximation on manifolds: General theory
math.NT
Abstract
We investigate the question of how well points on a nondegenerate -dimensional submanifold can be approximated by rationals also lying on , establishing an upper bound on the "intrinsic Dirichlet exponent" for . We show that relative to this exponent, the set of badly intrinsically approximable points is of full dimension and the set of very well intrinsically approximable points is of zero measure. Our bound on the intrinsic Dirichlet exponent is phrased in terms of an explicit function of and which does not seem to have appeared in the literature previously. It is shown to be optimal for several particular cases. The requirement that the rationals lie on distinguishes this question from the more common context of (ambient) Diophantine approximation on manifolds, and necessitates the development of new techniques. Our main tool is an analogue of the Simplex Lemma for rationals lying on which provides new insights on the local distribution of rational points on nondegenerate manifolds.
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Detailed mathematical audit
01Statements2 reported findingsCorrect
The full-dimension theorem for intrinsically badly approximable points on nondegenerate manifolds and the nullity theorem for intrinsically very well approximable points are correct.
Theorem 1.3Correct
Full dimension of intrinsic bad approximation
Pages 3–5 and Sections 3–4 · Theorem 1.3 · arXiv:1509.05439v3
The intrinsic simplex lemma places all sufficiently low-height rational points near a proper algebraic set on each small parameter ball. The algebraic-set game lets the player avoid that set at every scale, and nondegeneracy converts winning in parameter space to full Hausdorff dimension on the manifold.
Full paper, version 3 ↗Theorem 1.5Correct
Intrinsic very well approximable points are null
Pages 5–6 and Section 4 · Theorem 1.5 · arXiv:1509.05439v3
The simplex lemma and the manifold's local volume bounds give a summable cover at every exponent strictly better than the intrinsic Dirichlet exponent. Borel–Cantelli then makes the corresponding limsup set null in each coordinate chart, and a countable chart cover gives the global claim.
02Proofs2 reported findingsCorrect
The simplex lemma, algebraic-set winning construction, and measure-covering argument are correct and complete.
Lemma 4.1 and Theorem 4.3Correct and complete
The simplex bound supports the winning strategy
Section 4 · Lemma 4.1 and Theorem 4.3 · arXiv:1509.05439v3
The determinant estimate forces rational points below the height threshold into the announced algebraic hypersurface. At each game scale the player deletes its controlled neighborhood; nondegeneracy prevents the parameter chart from lying in that hypersurface and preserves a legal response.
Theorem 4.5Correct and complete
The null-set cover has a convergent total measure
Section 4 · Theorem 4.5 · arXiv:1509.05439v3
Rational points are grouped by height, the simplex lemma reduces each group to controlled neighborhoods, and local regularity bounds their measure. The improved exponent supplies a geometric decay factor, so summing over groups and charts proves nullity without an omitted endpoint case.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.