Effective rational approximation on spheres

TL;DR

This paper establishes an effective estimate for counting Diophantine approximants on spheres using advanced homogeneous dynamics techniques, improving understanding of approximation properties on spherical surfaces.

Contribution

It introduces new effective bounds for Diophantine approximation on spheres by leveraging recent advances in homogeneous dynamics and lattice theory.

Findings

Derived explicit counting estimates for approximants on spheres

Applied effective equidistribution results to Diophantine problems

Extended previous work with new bounds and techniques

Abstract

We prove an effective estimate for the counting function of Diophantine approximants on the sphere S$^n$. We use homogeneous dynamics on the space of orthogonal lattices, in particular effective equidistribution results and non-divergence estimates for the Siegel transform, developping on recent results of Alam-Ghosh and Kleinbock-Merrill.