that a vectorial function $F:\mathbb{F}_{p^{n}} \rightarrow \mathbb{F}_{p^{m}}$ is s-plateaued if and only if its graph is a partial geometric difference set with the parameters $(p^{n+m},p^n;p^{2n-m}-p^{n+s-m},p^{n+s}+p^{2n-m}-p^{n+s-m})$. Note that Theorem \[plateaued-PGDS\] is also valid for $m=1$, i.e. the case of p-ary functions. Since for an $s$-plateaued $p$-ary function $f:\mathbb{F}_{p^{n}}\rightarrow \mathbb{F}_{p}$, the function $bf(x)$ is $s$-plateaued for each $b\in \mathbb{F}_{p}^*$, we can consider $f$ as a vectorial $s$-plateaued function. The case $s=0$ is the case of vectorial bent functions and if we additionally have $m=n$, these vectorial functions are known as *planar* functions [@ColM].

Next we will investigate links between vectorial $s-$plateaued functions and partial geometric difference sets.

\[diff-plateauedprop\] Let $F:\mathbb{F}_{p^{n}}\rightarrow \mathbb{F}_{p^{m}}$ be a vectorial function. Then $F$ is $s$-plateaued if and only if $$\sum\limits_{a \in \mathbb{F}_{p^{n}}}|\{s \in \mathbb{F}_{p^{n}}\colon  y=F(s+x-a)-F(s)+F(a)\}|=\left \{\begin{array}{ll} \alpha & \mbox{if } y\neq F(x),\\
    \beta & \mbox{if } y=F(x)\\ \end{array} \right .$$

$$\begin{split}
    \delta((x,y))&=|\{((s_1,t_1),(s_2,t_2))\in G_F \times G_F \colon x=s_1-s_2, y=t_1-t_2=F(s_1)-F(s_2)\}|\\
    &=|\{s_2 \in \mathbb{F}_{p^{n}}\colon  y=F(s_2+x)-F(s_2)\}|\\
    \end{split}$$ So the criteria for PGDS is given by $$\sum\limits_{a \in \mathbb{F}_{p^{n}}}\delta((x-a,y-F(a)))=\left \{\begin{array}{ll} \alpha & \mbox{if } y\neq F(x),\\
    \beta & \mbox{if } y=F(x)\\ \end{array} \right .$$ and hence $$\sum\limits_{a \in \mathbb{F}_{p^{n}}}|\{s \in \mathbb{F}_{p^{n}}\colon  y=F(s+x-a)-F(s)+F(a)\}|=\left \{\begin{array}{ll} \alpha & \mbox{if } y\neq F(x),\\
    \beta & \mbox{if } 