Remarks about the C∞-closing lemma for 3-dimensional Reeb flows

We prove two refinements of theC∞-closing lemma for 3-dimensional Reeb flows, which was proved by the author as an application of spectral invariants ofEmbedded Contact Homology (ECH). Specifically, we prove the following two results: (i) for aC∞-generic contact form on any closed 3-manifold, the union of periodic Reeb orbits representing ECH homology classes is dense; (ii) a certain real-analytic version of the C∞-closing lemma for 3-dimensional Reeb flows. A few questions and conjectures related to these results are also discussed.