# Borel density for approximate lattices

## TL;DR

This paper extends classical density theorems to approximate lattices, showing that Zariski closures of approximate subgroups are close to algebraic subgroups, using stationary joinings in dynamical systems.

## Contribution

It introduces a novel extension of density theorems to approximate lattices and employs new dynamical systems techniques involving stationary joinings.

## Key findings

- Zariski closures of approximate subgroups are close to algebraic subgroups
- Extension of classical density theorems to approximate lattices
- Use of stationary joinings in the analysis

## Abstract

We extend classical density theorems of Borel and Dani--Shalom on lattices in semisimple, respectively solvable algebraic groups over local fields to approximate lattices. Our proofs are based on the observation that Zariski closures of approximate subgroups are close to algebraic subgroups. Our main tools are stationary joinings between the hull dynamical systems of discrete approximate subgroups and their Zariski closures.