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Lattice points in thickened parabolas and rational points near hypersurfaces

Alexander Smith

TL;DR

This work develops a dynamical framework to estimate how densely rational points can approximate nondegenerate hypersurfaces in Euclidean space. By introducing thickened parabolas via unipotent flows and applying Ratner-type theorems, it establishes heuristically sharp lower bounds for primitive lattice points near hypersurfaces, except for the rational quadric obstruction. It then localizes the problem to patches of quadric hypersurfaces, proving a uniform lower bound for lattice points in thickened parabola families and propagating this to global counts on hypersurfaces, with effective results in low dimensions. A Lie-theoretic analysis of the semisimple and unipotent components underpins the method, enabling a complete classification that supports the dynamical arguments. Overall, the paper advances precise lower bounds and effective results for rational points near hypersurfaces, highlighting the exceptional role of rational quadrics and connecting Diophantine approximation with homogeneous dynamics.

Abstract

Among the nondegenerate C^4 hypersurfaces M in R^n, we characterize the rational quadrics as the hypersurfaces that are the least well approximated by rational points. Given M other than a rational quadric, we prove a heuristically sharp lower bound for the number of rational points very near M, improving the sensitivity of prior results of Beresnevich and Huang. Our methods are dynamical, and rely on an application of Ratner's theorems to 1-parameter unipotent subgroups U of SL_n(R) such that u - Id has rank at most 2 for every u in U. As part of our work, we study the algebraic subgroups of SL_n(Q) whose collection of real points can contain such a subgroup.

Lattice points in thickened parabolas and rational points near hypersurfaces

TL;DR

This work develops a dynamical framework to estimate how densely rational points can approximate nondegenerate hypersurfaces in Euclidean space. By introducing thickened parabolas via unipotent flows and applying Ratner-type theorems, it establishes heuristically sharp lower bounds for primitive lattice points near hypersurfaces, except for the rational quadric obstruction. It then localizes the problem to patches of quadric hypersurfaces, proving a uniform lower bound for lattice points in thickened parabola families and propagating this to global counts on hypersurfaces, with effective results in low dimensions. A Lie-theoretic analysis of the semisimple and unipotent components underpins the method, enabling a complete classification that supports the dynamical arguments. Overall, the paper advances precise lower bounds and effective results for rational points near hypersurfaces, highlighting the exceptional role of rational quadrics and connecting Diophantine approximation with homogeneous dynamics.

Abstract

Among the nondegenerate C^4 hypersurfaces M in R^n, we characterize the rational quadrics as the hypersurfaces that are the least well approximated by rational points. Given M other than a rational quadric, we prove a heuristically sharp lower bound for the number of rational points very near M, improving the sensitivity of prior results of Beresnevich and Huang. Our methods are dynamical, and rely on an application of Ratner's theorems to 1-parameter unipotent subgroups U of SL_n(R) such that u - Id has rank at most 2 for every u in U. As part of our work, we study the algebraic subgroups of SL_n(Q) whose collection of real points can contain such a subgroup.

Paper Structure

This paper contains 24 sections, 27 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Rational points near hypersurfaces
  3. Thickened parabolas
  4. Diophantine approximation for systems of forms
  5. The layout of this paper
  6. Applying Ratner's theorems
  7. A version for functions
  8. The proof of Theorem \ref{['thm:uniform_parabola_func']}
  9. An effective version for $n = 3$
  10. Local approximation with quadrics
  11. Rational points near patches of quadric hypersurfaces
  12. The proof of Theorem \ref{['thm:main']}
  13. The proof of Theorems \ref{['thm:hypersurface_qual']}, \ref{['thm:main_eff']}, and \ref{['thm:planarcurves']}
  14. Semisimple Lie algebras
  15. The case of simple $\mathfrak{h}$
  16. ...and 9 more sections

Key Result

Theorem 1.1

Take $\mathcal{M}$ to be a $C^4$ hypersurface in $\mathbb{R}^n$. We assume that $\mathcal{M}$ is not contained in any hyperplane. Given $Q > 1$, define $\delta_{\mathcal{M}}(Q)$ to be the minimal $\delta > 0$ such that Then exactly one of the following holds:

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 2.2
  • Theorem 2.3
  • ...and 38 more