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On the Folklore set and Dirichlet spectrum for matrices

Mumtaz Hussain, Johannes Schleischitz, Benjamin Ward

TL;DR

This work advances the study of the Folklore set and Dirichlet spectrum for $m\times n$ matrices by providing constructive nonemptiness results and explicit lower bounds on the Hausdorff dimension of Folklore-type sets across a broad range of $(m,n)$ and norms. The authors develop a constructive method that first builds vectors of a prescribed Diophantine type in low dimensions and then lifts them to matrices via a block-structure approach inspired by Moshchevitin, enabling control over exact Dirichlet constants in several regimes. They establish that the Dirichlet spectrum contains right neighborhoods of 0 in many cases and derive dimension bounds that hold for arbitrary norms, complementing recent variational-principle-based results and recent full-spectrum findings. The results thereby enrich the understanding of uniform and ordinary Diophantine approximation for linear forms, with implications for the spectra and size of exceptional sets in high-dimensional approximation problems.

Abstract

We study the Folklore set of Dirichlet improvable matrices in $\mathbb R^{m\times n}$ which are neither singular nor badly approximable. We prove the non-emptiness for all positive integer pairs $m,n$ apart from $\{m,n\}=\{ 1,1\}$ and $\{m,n\}=\{ 2,3\}$ in a constructive manner. For a wide range of integer pairs $(m,n)$ we construct subsets of the Folklore set with an exact prescribed Dirichlet constant (in some right neighbourhood of $0$). This enables us to provide information on the Dirichlet Spectrum of matrices. The key technique of our construction is to build first vectors of a given Diophantine type, and then to show that most `liftings' to matrices will preserve this Diophantine type. This is a variant of a method introduced by Moshchevitin for uniform approximation. Our technique is often also applicable to arbitrary norms. As a corollary, we obtain lower bounds on the Hausdorff dimension of these sets. These statements complement previous results of the middle-named author (Selecta Math. 2023), Beresnevich et. al. (Adv. Math. 2023), and Das et. al. (Adv. Math. 2024).

On the Folklore set and Dirichlet spectrum for matrices

TL;DR

This work advances the study of the Folklore set and Dirichlet spectrum for matrices by providing constructive nonemptiness results and explicit lower bounds on the Hausdorff dimension of Folklore-type sets across a broad range of and norms. The authors develop a constructive method that first builds vectors of a prescribed Diophantine type in low dimensions and then lifts them to matrices via a block-structure approach inspired by Moshchevitin, enabling control over exact Dirichlet constants in several regimes. They establish that the Dirichlet spectrum contains right neighborhoods of 0 in many cases and derive dimension bounds that hold for arbitrary norms, complementing recent variational-principle-based results and recent full-spectrum findings. The results thereby enrich the understanding of uniform and ordinary Diophantine approximation for linear forms, with implications for the spectra and size of exceptional sets in high-dimensional approximation problems.

Abstract

We study the Folklore set of Dirichlet improvable matrices in which are neither singular nor badly approximable. We prove the non-emptiness for all positive integer pairs apart from and in a constructive manner. For a wide range of integer pairs we construct subsets of the Folklore set with an exact prescribed Dirichlet constant (in some right neighbourhood of ). This enables us to provide information on the Dirichlet Spectrum of matrices. The key technique of our construction is to build first vectors of a given Diophantine type, and then to show that most `liftings' to matrices will preserve this Diophantine type. This is a variant of a method introduced by Moshchevitin for uniform approximation. Our technique is often also applicable to arbitrary norms. As a corollary, we obtain lower bounds on the Hausdorff dimension of these sets. These statements complement previous results of the middle-named author (Selecta Math. 2023), Beresnevich et. al. (Adv. Math. 2023), and Das et. al. (Adv. Math. 2024).
Paper Structure (23 sections, 26 theorems, 260 equations)

This paper contains 23 sections, 26 theorems, 260 equations.

Table of Contents

  1. Uniform approximation for Systems of linear forms and arbitrary norms
  2. A note on recent advances
  3. On the Folklore set in the Dual approximation setting
  4. Maximum norm
  5. Arbitrary norms
  6. The Folklore set for a system of linear forms
  7. Two fundamental results for maximum norm
  8. The case $m/n\in [1/2,2]$
  9. The case $m/n\notin (1/2,2)$
  10. Brief outline of proofs
  11. Proof of Theorem \ref{['thm1']}
  12. Simplifying the problem
  13. Proof of Theorem \ref{['thm3']}
  14. Proof of Theorem \ref{['jia']}
  15. Outline
  16. ...and 8 more sections

Key Result

Lemma 1.1

Given $m,n$, there exists $\delta=\delta_{m,n}>0$ so that if $\Omega \in FS_{m,n}(\|\cdot\|_{1},\|\cdot\|_{2}, c)$ for some $0<c<\delta$, then its transpose $\Omega^T$ satisfies $\Omega^{T} \in FS_{n,m}(\|\cdot\|_{2},\|\cdot\|_{1})$.

Theorems & Definitions (62)

  • Lemma 1.1
  • Theorem 2.1
  • Remark 1
  • Remark 2
  • Corollary 2.2
  • Remark 3
  • Theorem 2.3
  • Remark 4
  • Remark 5
  • Remark 6
  • ...and 52 more