Equidistribution, covering radius, and Diophantine approximation for rational points on the sphere

TL;DR

This work analyzes the distribution of rational points on the sphere and their small-scale equidistribution properties via spherical harmonics and theta-series methods. In particular, it proves strong shrinking-radius equidistribution bounds in general and an optimal $R\gg n^{-1/4+o(1)}$ range for $S^2$ using Hecke operators from quaternion algebras, along with a variance-based almost-everywhere improvement to $R\gg n^{-1/2+o(1)}$. The paper connects these results to covering radii and intrinsic Diophantine approximation, detailing the covering exponents $K((\Omega_T))=2$, $K_\mu((\Omega_T))=1$, and dimension-dependent bounds for $\Omega_n$, and it advances Linnik-type questions by showing a primitive solution with $|z|=O(n^{1/3+\epsilon})$ for odd squares. These insights illuminate the limits and potential optimality of small-scale equidistribution in dimension two, with broader implications for Diophantine approximation on spheres and related arithmetic geometry.

Abstract

We consider rational points on the sphere and investigate their equidistribution in shrinking spherical caps. For the two-dimensional sphere, we leverage Hecke operators to obtain a significantly improved small-scale equidistribution bound, and discuss connections to the covering radius problem, intrinsic Diophantine approximation, and Linnik's conjecture on sums of two squares and a mini-square.

Figures (3)

• Figure 1: $\Omega_n\subset S^2$ for $n=101,201,401,701$
• Figure 2: $\Omega_T\subset S^2$ for $T=1,20,40,100$
• Figure 3: $\Omega_n\subset S^2$ for $n=163,171,269,287$

Theorems & Definitions (17)

• Theorem 1.1: Small-Scale Equidistribution
• Theorem 1.2: Small-Scale Equidistribution on $S^2$
• Conjecture 1.3: Linnik
• Definition 1.4: Generic Covering Exponent
• Proposition 1.5: Covering Exponents for $(\Omega_T)$
• Corollary 1.6
• Theorem 1.7
• Lemma 2.1
• proof
• Proposition 2.2