Vectorial Boolean functions on distance one from APN functions

The metric properties of the class of vectorial Boolean functions are studied. A vectorial Boolean function $F$ in $n$ variables is called a differential $\delta$-uniform function if the equation $F(x)\oplus F(x\oplus a)=b$ has at most $\delta$ solutions for any vectors $a,b$, where $a\neq0$. In particular, if it is true for $\delta=2$, then the function $f$ is called APN. The distance between vectorial Boolean functions $F$ and $G$ is the cardinality of the set $\{x\in\mathbb Z_2^n\colon F(x)\neq G(x)\}$. It is proved that there are only differential $4$-uniform functions which are on the distance 1 from an APN function.