Motivated by the Jacobian problem, this article is concerned with the density of the image set $F(\mathbb Z^n)$ of polynomial maps $F\in\mathbb Z[X_1,\dots,X_n]^n$ with $\det DF\equiv 1$. It is shown that if such a map $F$ is not invertible, its image set $F(\mathbb Z^n)$ must be very thin in the lattice $\mathbb Z^n$: (1) for almost all lines $l$ in $\mathbb Z^n$ the numbers $\#(F^{−1}(l)\cap \mathbb Z^n)$ are uniformly bounded; (2) $\#\{z\in F(\mathbb Z^n)\,:\,|z_i|\geq B\} \ll B_{n−1}$ as $B\to +\infty$, where the implicit constants depend on $F$.