Local distribution of rational points in flag varieties
Zhizhong Huang, Nicolas de Saxcé
TL;DR
The paper analyzes how rational points on flag varieties approximate real points when heights grow and distances shrink, proving that zoomed, height-filtered rational points equidistribute under generic or suitably nondegenerate conditions. It develops a comprehensive framework combining height theory on X=G/P, reduction theory, and homogeneous dynamics to study both global counts N(X,H) and local zooming measures μ_{x,τ,t}, establishing asymptotics linked to Tamagawa measures and Peyre constants. For rank-one flag varieties with abelian unipotent radical, it proves explicit local distribution results and extends these to general flag varieties via Carnot–Carathéodory geometry and effective counting in low lattices. The work thus provides a dynamical approach to Diophantine approximation on flag varieties, connects local zooming behavior to global height asymptotics, and clarifies how Peyre-type constants govern both counting and local distribution, with open questions about uniformity, higher rank, and optimal zoom ranges.
Abstract
Given a flag variety $X$ defined over $\mathbb{Q}$ and a point $x$ in $X(\mathbb{R})$, we study approximations to $x$ by points $v$ in $X(\mathbb{Q})$, and show that, with an appropriate rescaling, those approximations equidistribute when $x$ is chosen randomly according to a Lebesgue measure on $X(\mathbb{R})$, or when $x$ is defined over $\mathbb{Q}$ and satisfies some non-degeneracy condition.