Constructing exact Lagrangian immersions with few double points

We establish an $h$-principle for exact Lagrangian immersions with transverse self-intersections and the minimal, or near-minimal number of double points. One corollary of our result is that any orientable closed 3-manifold admits an exact Lagrangian immersion into standard symplectic 6-space ${\mathbb{R}}_{\text{st}}^6$ with exactly one transverse double point. Our construction also yields a Lagrangian embedding $S^1\times S^2\to {\mathbb{R}}_{\text{st}}^6$ with vanishing Maslov class.