 Large Freiman-homomorphic restriction from bounded derivative images

Suppose every derivative $x\mapsto \varphi (x+d)-\varphi (x)$ on ${\mathbb{F}}_p^n$ takes at most $C$ values. Then $\varphi$ restricts to a Freiman homomorphism on a set of density at least $p^{-1}{C}^{-5}$. The proof bounds the second-difference image by $C^5$ and chooses a random subspace avoiding its nonzero elements.