# Approximation diophantienne et distribution locale sur une surface torique

## TL;DR

This paper investigates Diophantine approximation and local distribution of rational points on a specific toric surface, revealing that optimal approximations occur along nodal curves and reducing the problem to quadratic points on the projective line.

## Contribution

It provides a detailed analysis of approximation behavior on a blown-up toric surface, linking it to the distribution of quadratic points on the projective line, a novel geometric approach.

## Key findings

- Optimal approximations are achieved via nodal curves.
- The problem reduces to the distribution of quadratic points on the projective line.
- Outside a Zariski closed subset, the approximation behavior is characterized.

## Abstract

In this article we study Diophantine approximation and local distribution of a rational point on a toric surface obtained as a blow-up of $\mathbb{P}^1\times\mathbb{P}^1$. It turns out that outside a Zariski closed subset the best approximations are achieved through a family of nodal curves. Hence the investigation is reduced to the question of local distribution of a quadratic point on the projective line.