Optimal quantitative weak approximation for projective quadrics

TL;DR

This paper provides asymptotic formulas for counting rational points on high-dimensional smooth projective quadrics within shrinking adelic neighborhoods, advancing the understanding of weak approximation and equidistribution in algebraic geometry.

Contribution

It introduces the first quantitative weak approximation results for projective quadrics, establishing optimal growth rates for adelic neighborhoods.

Findings

Derived asymptotic formulas for rational points on quadrics.

Identified the optimal size of adelic neighborhoods for equidistribution.

Enhanced understanding of weak approximation in higher-dimensional quadrics.

Abstract

We derive asymptotic formulas for the number of rational points on a smooth projective quadratic hypersurface of dimension at least three inside of a shrinking adelic open neighbourhood. This is a quantitative version of weak approximation for quadrics and allows us to deduce the best growth rate of the size of such an adelic neighbourhood for which equidistribution is preserved.