We consider the dynamics of a Reeb vector field on a three-manifold (which is closely related to the dynamics of area-preserving surface diffeomorphisms). A "closing lemma" is a statement asserting that one can make a small perturbation of a vector field (or diffeomorphism) to arrange that there is a periodic orbit passing through a given nonempty open set. In 2015, Irie proved a C-infinity closing lemma for Reeb vector fields in three dimensions. The goal of this talk is to describe a new approach to proving "quantitative" closing lemmas, which gives upper bounds on how much one needs to perturb in order to obtain a periodic orbit with a given upper bound on the period. A key technical tool will be "elementary spectral invariants" which measure the minimum energy of holomorphic curves in four dimensions.