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Extremal affine subspaces and Khintchine-Jarník type theorems

Jing-Jing Huang

Abstract

We prove a conjecture of Kleinbock which gives a clear-cut classification of all extremal affine subspaces of $\mathbb{R}^n$. We also give an essentially complete classification of all Khintchine type affine subspaces, except for some boundary cases within two logarithmic scales. More general Jarník type theorems are proved as well, sometimes without the monotonicity of the approximation function. These results follow as consequences of our novel estimates for the number of rational points close to an affine subspace in terms of diophantine properties of its defining matrix. Our main tool is the multidimensional large sieve inequality and its dual form.

Extremal affine subspaces and Khintchine-Jarník type theorems

Abstract

We prove a conjecture of Kleinbock which gives a clear-cut classification of all extremal affine subspaces of . We also give an essentially complete classification of all Khintchine type affine subspaces, except for some boundary cases within two logarithmic scales. More general Jarník type theorems are proved as well, sometimes without the monotonicity of the approximation function. These results follow as consequences of our novel estimates for the number of rational points close to an affine subspace in terms of diophantine properties of its defining matrix. Our main tool is the multidimensional large sieve inequality and its dual form.
Paper Structure (35 sections, 33 theorems, 224 equations)

This paper contains 35 sections, 33 theorems, 224 equations.

Table of Contents

  1. Diophantine exponents and extremal manifolds
  2. Dirichlet's theorem and diophantine exponents of matrices
  3. Extremal manifolds
  4. Diophantine exponents of affine subspaces
  5. Khintchine-Jarník type theorems
  6. Khintchine type manifolds
  7. Khintchine type affine subspaces
  8. Strong Khintchine type manifolds
  9. Multiplicative approximation on planar lines
  10. Jarník type manifolds
  11. Hausdorff dimension of very well approximable vectors
  12. Counting rational points near affine subspaces
  13. Main counting results
  14. The upper bound estimates
  15. Some preliminary lemmata
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $\mathcal{L}$ be an affine subspace parametrized by $A$ as in AffineSubspace. Then we have the following equivalence:

Theorems & Definitions (61)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.2
  • Corollary 1.3
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.4
  • ...and 51 more