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Rational Points in Hyperbolic Regions and Multiplicative Diophantine Approximation on Manifolds

Sam Chow, Rajula Srivastava, Niclas Technau, Han Yu

TL;DR

This work develops a comprehensive convergence theory for multiplicative Diophantine approximation on smooth non-degenerate manifolds and certain affine subspaces under a generic Diophantine condition. The authors reduce curved cases to non-degenerate curves via a fibring lemma and treat the flat case using quantitative non-divergence estimates for spaces of lattices, together with detailed harmonic-analytic counting in anisotropic regions. The main results establish Gallagher-type convergence for the ambient Lebesgue measure on manifolds and for affine subspaces satisfying Hypothesis D, resolving longstanding questions and sharpening extremality results. The methodology combines target-hitting arguments, dyadic decompositions, Poisson summation, oscillatory integral bounds, and robust non-divergence machinery, with careful accounting of parameter dependencies across the flat/curved dichotomy. These results advance our understanding of rational points near manifolds and provide a versatile framework for multiplicative Diophantine problems in non-flat geometric settings.

Abstract

We establish the convergence theory of multiplicative Diophantine approximation for all non-degenerate, smooth manifolds. We also settle said convergence theory for all affine subspaces satisfying a highly generic and essentially optimal Diophantine condition. This answers a question of Beresnevich and Velani from 2005, while simultaneously sharpening results of Kleinbock and Margulis on the strong extremality of non-degenerate manifolds, and of Kleinbock on the strong extremality of affine subspaces.

Rational Points in Hyperbolic Regions and Multiplicative Diophantine Approximation on Manifolds

TL;DR

This work develops a comprehensive convergence theory for multiplicative Diophantine approximation on smooth non-degenerate manifolds and certain affine subspaces under a generic Diophantine condition. The authors reduce curved cases to non-degenerate curves via a fibring lemma and treat the flat case using quantitative non-divergence estimates for spaces of lattices, together with detailed harmonic-analytic counting in anisotropic regions. The main results establish Gallagher-type convergence for the ambient Lebesgue measure on manifolds and for affine subspaces satisfying Hypothesis D, resolving longstanding questions and sharpening extremality results. The methodology combines target-hitting arguments, dyadic decompositions, Poisson summation, oscillatory integral bounds, and robust non-divergence machinery, with careful accounting of parameter dependencies across the flat/curved dichotomy. These results advance our understanding of rational points near manifolds and provide a versatile framework for multiplicative Diophantine problems in non-flat geometric settings.

Abstract

We establish the convergence theory of multiplicative Diophantine approximation for all non-degenerate, smooth manifolds. We also settle said convergence theory for all affine subspaces satisfying a highly generic and essentially optimal Diophantine condition. This answers a question of Beresnevich and Velani from 2005, while simultaneously sharpening results of Kleinbock and Margulis on the strong extremality of non-degenerate manifolds, and of Kleinbock on the strong extremality of affine subspaces.
Paper Structure (37 sections, 22 theorems, 266 equations)

This paper contains 37 sections, 22 theorems, 266 equations.

Table of Contents

  1. Introduction and Main Results
  2. Introduction
  3. Simultaneous and multiplicative approximation
  4. Diophantine approximation on manifolds
  5. Main results
  6. Further remarks and open problems
  7. Organisation
  8. Notation
  9. Outline of the proofs
  10. A high-level overview
  11. The flat case, i.e. Theorem \ref{['FlatConv']}
  12. The curved case, i.e. Theorem \ref{['thm: Gallagher conv']}
  13. The flat setting
  14. Diophantine hypothesis
  15. Equidistribution on tori
  16. ...and 22 more sections

Key Result

Theorem 1.5

Let $\mathcal{M}\subseteq{\mathbb R}^{n}$ be a smooth, non-degenerate manifold of dimension $d\geqslant 1$. Then the natural Lebesgue measure on $\mathcal{M}$ is convergent Gallagher.

Theorems & Definitions (60)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Conjecture 1.9
  • Remark 1.10
  • ...and 50 more