Weak approximation of symmetric products and norm varieties

Let k be a number field. For a variety X over k that satisfies weak approximation with Brauer-Manin obstruction, we study the same property for smooth projective models of its symmetric products. Based on the same method, we also explore the property of weak approximation with Brauer-Manin obstruction for norm varieties.
Comment: 9 pages; improved theorem 1.3