  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 $\mathbb{R} \times \mathbb{Q}_{p_1} \times \cdots
\times \mathbb{Q}_{p_r}$ by elements of the diagonal embedding of $\mathbb{Q}$
into this space.
