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Badly approximable grids and k-divergent lattices

Nikolay Moshchevitin, Anurag Rao, Uri Shapira

Abstract

For an m by n real matrix A, we investigate the set of badly approximable targets for A as a subset of the m-torus. It is well known that this set is large in the sense that it is dense and has full Hausdorff dimension. We investigate the relationship between its measure and Diophantine properties of A. On the one hand, we give the first examples of a non-singular matrix A such that the set of badly approximable targets has full measure with respect to some non-trivial algebraic measure on the torus. For this, we use transference theorems due to Jarnik and Khintchine, and the parametric geometry of numbers in the sense of Roy. On the other hand, we give a novel Diophantine condition on A that slightly strengthens non-singularity, and show that under the assumption that A satisfies this condition, the set of badly approximable targets is a null-set with respect to any non-trivial algebraic measure on the torus. For this we use naive homogeneous dynamics, harmonic analysis, and a novel concept we refer to as mixing convergence of measures.

Badly approximable grids and k-divergent lattices

Abstract

For an m by n real matrix A, we investigate the set of badly approximable targets for A as a subset of the m-torus. It is well known that this set is large in the sense that it is dense and has full Hausdorff dimension. We investigate the relationship between its measure and Diophantine properties of A. On the one hand, we give the first examples of a non-singular matrix A such that the set of badly approximable targets has full measure with respect to some non-trivial algebraic measure on the torus. For this, we use transference theorems due to Jarnik and Khintchine, and the parametric geometry of numbers in the sense of Roy. On the other hand, we give a novel Diophantine condition on A that slightly strengthens non-singularity, and show that under the assumption that A satisfies this condition, the set of badly approximable targets is a null-set with respect to any non-trivial algebraic measure on the torus. For this we use naive homogeneous dynamics, harmonic analysis, and a novel concept we refer to as mixing convergence of measures.
Paper Structure (23 sections, 46 theorems, 228 equations, 2 figures)

This paper contains 23 sections, 46 theorems, 228 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The origins of this work
  3. Inhomogeneous Diophantine approximation
  4. A few open problems
  5. From Inhomogeneous approximations to value sets of grids
  6. Euclidean space
  7. The space of lattices
  8. The space of grids
  9. The projection from grids to lattices and its fibers
  10. Subtori, Haar measures, and algebraic measures
  11. Value sets and inheritance
  12. An intriguing example
  13. The mistake in GDV
  14. Generalities about measures
  15. Borel measures and weak* convergence
  16. ...and 8 more sections

Key Result

Theorem 1.1

For any $A \in \operatorname{Mat}_{m\times n}(\mathbb{R})$, there exists an $\eta \in {\mathbb{R}}^m$ for which

Figures (2)

  • Figure 1: Here is a graph of $\Psi$ to remember the notation.
  • Figure 2: The orange, blue are olive graphs are called $P_1$, $P_2$, $P_3$ respectively.

Theorems & Definitions (118)

  • Theorem 1.1: Theorem X, Chapter IV CasselsDA
  • Theorem 1.2: Theorem 1.4 in ETcrelle or Theorem 1 in Moshchevitin-badforms
  • Definition 1.4
  • Definition 1.5: Singularity
  • Theorem 1.6: GDV
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.10: Theorem 2.14 in Dadivergent
  • Definition 1.11: Asymptotic accumulation points
  • ...and 108 more