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Uniform Diophantine approximation with restrictions via total density of collections of subspaces

Leo Hong, Dmitry Kleinbock, Vasiliy Neckrasov

Abstract

In 1926 Khintchine introduced a topological argument proving the existence of uncountably many nontrivial singular linear forms of $n \geq 2$ variables. Throughout the years, this argument has been extensively modified and generalized. Most recently, Kleinbock et al. (2025) introduced a general framework of Diophantine systems and showed that a certain topological property called total density implies a far-reaching generalization of Khintchine's result. We describe a way to establish total density for a variety of Diophantine systems, and thus prove that the sets of singular objects are uncountable and dense in a wide range of set-ups in Diophantine approximation. As a special case, we establish such a result for inhomogeneous approximation, proving the existence of uncountably many singular systems of affine forms with a fixed translation part. One can also consider approximation with prime denominators, or more generally, approximation under some strong restrictions on numerators and denominators.

Uniform Diophantine approximation with restrictions via total density of collections of subspaces

Abstract

In 1926 Khintchine introduced a topological argument proving the existence of uncountably many nontrivial singular linear forms of variables. Throughout the years, this argument has been extensively modified and generalized. Most recently, Kleinbock et al. (2025) introduced a general framework of Diophantine systems and showed that a certain topological property called total density implies a far-reaching generalization of Khintchine's result. We describe a way to establish total density for a variety of Diophantine systems, and thus prove that the sets of singular objects are uncountable and dense in a wide range of set-ups in Diophantine approximation. As a special case, we establish such a result for inhomogeneous approximation, proving the existence of uncountably many singular systems of affine forms with a fixed translation part. One can also consider approximation with prime denominators, or more generally, approximation under some strong restrictions on numerators and denominators.

Paper Structure

This paper contains 28 sections, 37 theorems, 196 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The Khintchine--Jarnik Theorem and total density
  3. Uniform approximation with restrictions
  4. Inhomogeneous Diophantine approximation
  5. The structure of the paper
  6. Acknowledgments
  7. Main results
  8. Two general theorems
  9. Proof of Theorem \ref{['not_on_line']}
  10. Other applications and examples
  11. Two sufficient conditions for density
  12. Some general conditions implying total density
  13. Proof of the main results
  14. Proof of Theorem \ref{['any_k_theorem']}
  15. Proof of Theorem \ref{['ch_theorem']}
  16. ...and 13 more sections

Key Result

Theorem A

Let $m,n\in\mathbb{N}$.

Figures (7)

  • Figure 1: The set $Q$ in Case (i)
  • Figure 2: The set $Q$ in Case (ii)
  • Figure 3: The set $Q$ in Case (iii)
  • Figure 4: The set $Q$ in Case (iv)
  • Figure 5: The set $Q$ in Case (v)
  • ...and 2 more figures

Theorems & Definitions (83)

  • Definition 1.1
  • Theorem A
  • Definition 1.2
  • Definition 1.3
  • Theorem B
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Corollary 1.8
  • ...and 73 more