Diagonal approximation in completions of [formula omitted].

We prove analogues of some classical results from Diophantine approximation and metric number theory (namely Dirichlet's theorem and the Duffin–Schaeffer theorem) in the setting of diagonal Diophantine approximation, i.e. approximating elements of R × Q p 1 × ⋯ × Q p r by elements of the diagonal embedding of Q into this space. [ABSTRACT FROM AUTHOR]