Proof of Gromov's theorem on homogeneous nilpotent approximation for vector fields of class $C^1$

The article is devoted to the asymptotic properties of the vector fields $\widetilde X^g_i$, $i=1,\dots,N$, $\theta_g$-connected with $C^1$-smooth basis vector fields $\{X_i\}_{i=1,\dots,N}$ satisfying condition $(+\deg)$. We prove a theorem of Gromov on the homogeneous nilpotent approximation for vector fields of class $C^1$. Nontrivial examples are constructed of quasimetrics induced by vector fields $\{X_i\}_{i=1,\dots,N}$.