arXiv:math/9808057v2
Badly approximable systems of affine forms
math.NTmath.DS11J2011J8357S25
Abstract
We prove an inhomogeneous analogue of W. M. Schmidt's (1969) theorem on Hausdorff dimension of the set of badly approximable systems of linear forms. The proof is based on ideas and methods from the theory of dynamical systems, in particular, on abundance of bounded orbits of mixing flows on homogeneous spaces of Lie groups.
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Detailed mathematical audit
01Statements2 reported findingsCorrect
The thickness theorem for badly approximable systems of affine forms and the general bounded-orbit avoidance theorem from which it follows are correct.
Theorem 1.5Correct
Badly approximable affine systems form a thick set
Pages 3–4 and Section 6 · Theorem 1.5 · arXiv:math/9808057v2
Dani's correspondence identifies bad affine approximation with boundedness of the appropriate homogeneous trajectory. Applying the dimension construction to the expanding horospherical parameter space gives full Hausdorff dimension in every open set, which is exactly thickness.
Full paper, version 2 ↗Theorems 1.6 and 1.7Correct
Bounded trajectories avoiding null invariant sets
Pages 4–6 and Sections 2–5 · Theorems 1.6–1.7 · arXiv:math/9808057v2
Exponential mixing gives uniform distribution of most descendants in each tessellation cell, while the null invariant set can be covered by cells of arbitrarily small retained proportion. The nested construction therefore has dimension approaching the full unstable dimension and every selected orbit remains bounded and avoids the target.
02Proofs2 reported findingsCorrect
The mixing estimate, tessellation construction, Frostman dimension bound, and affine Dani correspondence are correct and complete.
Sections 2–5Correct and complete
The nested-cell construction has full limiting dimension
Sections 2–5 · arXiv:math/9808057v2
At each generation mixing controls the number of cells entering the forbidden neighborhood, leaving a uniform proportion of children. The constructed probability measure satisfies the required ball estimate; letting the scale and forbidden-neighborhood parameters tend in the stated order makes its dimension arbitrarily close to the ambient unstable dimension.
Section 6Correct and complete
The affine-form correspondence preserves the bad-approximation constant
Section 6 · proof of Theorem 1.5 · arXiv:math/9808057v2
The block-unipotent lattice records the affine errors and denominators as coordinates of a lattice vector. Uniform avoidance of the cusp is equivalent, via Mahler compactness, to a positive lower bound for the affine products, so Theorem 1.6 applies without an unhandled rational case.
03Novelty0 reported findingsNo non-novelty findings
No non-novelty findings.