  Let $I$ be an inner function in $\mathcal{D} = B_{n_1}\times B_{n_2}\cdots
\times B_{n_k}$, where $B_n$ denotes the open unit ball of $\mathbb{C}^n$,
$n\ge 1$. We construct dominant sets for the space $H^2 \ominus I H^2$, where
$H^2 = H^2(\mathcal{D})$ denotes the standard Hardy space.
