Let
${\mathbb{F}}_{q}$ be a finite
field of order
$q$.
We prove that if
$d\ge 2$
is even and
$E\subset {\mathbb{F}}_{q}^{d}$
with
$\leftE\right\ge 9{q}^{d\u22152}$
then
$${\mathbb{F}}_{q}=\frac{\Delta \left(E\right)}{\Delta \left(E\right)}=\left\{\frac{a}{b}:a\in \Delta \left(E\right),b\in \Delta \left(E\right)\setminus \left\{0\right\}\right\},$$
where
$$\Delta \left(E\right)=\left\{\parallel xy\parallel :x,y\in E\right\},\phantom{\rule{1em}{0ex}}\parallel x\parallel ={x}_{1}^{2}+{x}_{2}^{2}+\cdots +{x}_{d}^{2}.$$
If the dimension
$d$
is odd and
$E\subset {\mathbb{F}}_{q}^{d}$
with
$\leftE\right\ge 6{q}^{d\u22152}$,
then
$$\left\{0\right\}\cup {\mathbb{F}}_{q}^{+}\subset \frac{\Delta \left(E\right)}{\Delta \left(E\right)},$$
where
${\mathbb{F}}_{q}^{+}$
denotes the set of nonzero quadratic residues in
${\mathbb{F}}_{q}$. Both
results are, in general, best possible, including the conclusion about the nonzero
quadratic residues in odd dimensions.
