arXiv:1601.05990v2
A note on badly approximable linear forms on manifolds
math.NT
Abstract
This paper is motivated by Davenport's problem and the subsequent work regarding badly approximable points in submanifolds of a Euclidian space. We study the problem in the area of twisted Diophantine approximation and present two different approaches. The first approach shows that, under a certain restriction, any countable intersection of the sets of weighted badly approximable points on any non-degenerate C^1 submanifold of R^n has full dimension. In the second approach we introduce the property of isotropically winning and show that the sets of weighted badly approximable points are isotropically winning under the same restriction as above.
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01Statements2 reported findingsCorrect
The winning and full-dimension conclusions on nondegenerate manifolds, and the isotropic-winning conclusion on affine subspaces, are supported under the stated weighted homogeneous badness assumption.
Theorem 2.1 and Corollary 2.2Correct
Twisted bad sets are winning on curves and full-dimensional on manifolds
Sections 2-3 · Theorem 2.1, Theorem 3.1, and Corollary 2.2 · arXiv:1601.05990v2
Projecting an everywhere nondegenerate curve to a coordinate with maximal weight is bi-Lipschitz. The game argument separates every scale's resonant points using the homogeneous badness constant, giving a one-quarter winning projected set. Countable intersection and the stated fibering argument then yield full intrinsic dimension on a nondegenerate manifold.
Theorem 2.4Correct
The twisted set is isotropically winning
Sections 2 and 4 · Theorem 2.4 and Facts A-D · arXiv:1601.05990v2
For every affine subspace, the proof constructs dual integer vectors whose orthogonal projections onto its direction have lacunary Euclidean norms. Avoiding the corresponding affine resonances is one-half winning, and the scalar-product transference estimate places that avoidance set inside the desired twisted badly approximable set.
02Proofs2 reported findingsCorrect
Both proof strategies control all resonances at the relevant scales; the projected lacunarity required for arbitrary affine slices is established quantitatively.
Theorem 3.1Correct and complete
The curve-game separation estimate is complete
Section 3 · Theorem 3.1 and scale classes · arXiv:1601.05990v2
Two resonances of the same scale inside Bob's interval would subtract to a nonzero homogeneous approximation contradicting . Hence Alice avoids the single dangerous subinterval at that stage; the limiting point retains a uniform weighted lower bound for every nonzero integer vector.
Lemma 4.1Correct
The projected dual vectors satisfy the required lacunarity
Sections 4.4-4.6 · Lemma 4.1 and angle estimate (30) · arXiv:1601.05990v2
Minkowski's theorem selects a dual vector in each anisotropic parallelepiped. The chosen thin directions keep it within the fixed angle cone, and geometrically spaced parameters make its projected norm grow by at least two. The cosine comparison explicitly transfers ambient growth to every selected affine direction.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.