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Effective estimate and Central Limit Theorem for Diophantine approximation on spheres

Zouhair Ouaggag

TL;DR

The paper advances intrinsic Diophantine approximation on spheres by proving an effective counting formula with a square-root error term and a Central Limit Theorem for the counting function N_{T,c}(α). It leverages the Dani correspondence to translate counting into ergodic averages of light-cone Siegel transforms on the space of orthogonal lattices, then develops a robust framework of truncation, smoothing, and higher-order equidistribution via Eisenstein-series techniques. A key technical achievement is the effective, multi-order equidistribution on K-orbits pushed by a_t, together with precise variance formulas derived from second-moment analysis, enabling both almost-everywhere error bounds and Gaussian limit laws. The results deepen the connection between homogeneous dynamics and intrinsic Diophantine questions on manifolds, with explicit constants and variance expressions enabling precise asymptotics and fluctuation theory for rational approximations on spheres.

Abstract

We study the counting function of rational approximations with given bounds on the denominator and satisfying the critical Dirichlet exponent on the sphere $S^d$, $d\geq 3$. We give an effective estimate for this counting function, with an error term of square root order, analogous to the optimal estimate in the Euclidean setting. We also show that the counting function has vanishing third and higher correlations and derive a Central Limit Theorem describing its fluctuations. We prove these results using arguments from homogeneous dynamics on the space of orthogonal lattices, in particular effective multiple equidistribution of all orders, which we establish for spherical averages and which could be useful for other applications.

Effective estimate and Central Limit Theorem for Diophantine approximation on spheres

TL;DR

The paper advances intrinsic Diophantine approximation on spheres by proving an effective counting formula with a square-root error term and a Central Limit Theorem for the counting function N_{T,c}(α). It leverages the Dani correspondence to translate counting into ergodic averages of light-cone Siegel transforms on the space of orthogonal lattices, then develops a robust framework of truncation, smoothing, and higher-order equidistribution via Eisenstein-series techniques. A key technical achievement is the effective, multi-order equidistribution on K-orbits pushed by a_t, together with precise variance formulas derived from second-moment analysis, enabling both almost-everywhere error bounds and Gaussian limit laws. The results deepen the connection between homogeneous dynamics and intrinsic Diophantine questions on manifolds, with explicit constants and variance expressions enabling precise asymptotics and fluctuation theory for rational approximations on spheres.

Abstract

We study the counting function of rational approximations with given bounds on the denominator and satisfying the critical Dirichlet exponent on the sphere , . We give an effective estimate for this counting function, with an error term of square root order, analogous to the optimal estimate in the Euclidean setting. We also show that the counting function has vanishing third and higher correlations and derive a Central Limit Theorem describing its fluctuations. We prove these results using arguments from homogeneous dynamics on the space of orthogonal lattices, in particular effective multiple equidistribution of all orders, which we establish for spherical averages and which could be useful for other applications.
Paper Structure (36 sections, 41 theorems, 377 equations)

This paper contains 36 sections, 41 theorems, 377 equations.

Table of Contents

  1. Introduction and main results
  2. First main result: Effective estimate for the counting function $\mathsf{N}_{T,c}$
  3. Outline of Section \ref{['sec:counting']}
  4. Second main result: Limit distribution of the counting function $\mathsf{N}_{T,c}$
  5. Outline of Section \ref{['sec:clt counting']}
  6. Acknowledgment
  7. Effective estimate for counting Diophantine approximations on spheres
  8. Diophantine approximation and dynamics on the space of lattices
  9. Tesselation of $E_{T,c}$
  10. The counting function $\mathsf{N}_{T,c}$ as ergodic averages on the light-cone
  11. Estimates on higher order correlations
  12. Siegel transform and approximation of the counting function
  13. Siegel transform
  14. Non-divergence estimates
  15. Truncated Siegel transform
  16. ...and 21 more sections

Key Result

Theorem 1.1

There exists a constant $C \geq 1$ such that for every $\alpha \in S^n$, there exist infinitely many rationals $\frac{p}{q} \in S^d$ such that

Theorems & Definitions (63)

  • Theorem 1.1: kleinbock2013rational, Theorem 1.1.
  • Theorem 1.2: kleinbock2013rational, Theorem 1.3.
  • Theorem 1.3: alam2020quantitative, Theorem 1.2.
  • Theorem 1.4: secondmomentKY23, Theorem 1.9.
  • Remark 1.5
  • Remark 1.6
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 53 more