  In this paper, we study the gap rigidity phenomenon for proper holomorphic
maps between balls of different dimension. We show that any $F\in
prop_3({\mathbb{B}}^n, {\mathbb{B}} ^N)$, with $3n<N\leq 4n-7$ and $n\geq 7$,
is equivalent to a map of the form $(G,0)$ with $G\in Rat
({\mathbb{B}}^n,{\mathbb{B}}^{3n})$. The main ingredients for the proof of our
main theorem are the normal form obtained by Huang-Ji-Xu and a lemma of the
first author.
