Counting rational approximations on rank one flag varieties

TL;DR

This paper studies how well real points on rank one flag varieties, including Grassmannians and quadrics, can be approximated by rational points, using advanced techniques from dynamics and geometry of numbers.

Contribution

It provides new counting results for rational approximations on rank one flag varieties, extending previous work to a broader class of geometric spaces.

Findings

Quantitative bounds on rational approximations

Application to Grassmann varieties and quadrics

Use of exponential mixing and geometry of numbers

Abstract

On a generalized flag variety of rank one, we count rational approximations to a real point chosen randomly according to the Riemannian volume. In particular, our results apply to Grassmann varieties and quadric hypersurfaces. The proof uses exponential mixing in the space of lattices and tools from geometry of numbers.