  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.
