Let
${\mathbb{\mathbb{F}}}_{q}$ be an arbitrary
finite field, and
$\mathcal{\mathcal{E}}$
be a point set in
${\mathbb{\mathbb{F}}}_{q}^{d}$.
Let
$\Delta \left(\mathcal{\mathcal{E}}\right)$
be the set of distances determined by pairs of points in
$\mathcal{\mathcal{E}}$.
Using Kloosterman sums, Iosevich and Rudnev (2007) proved that if
$\left\mathcal{\mathcal{E}}\right\ge 4{q}^{\left(d+1\right)\u22152}$ then
$\Delta \left(\mathcal{\mathcal{E}}\right)={\mathbb{\mathbb{F}}}_{q}$. In
general, this result is sharp in odddimensional spaces over arbitrary finite fields.
We use the pointplane incidence bound due to Rudnev to prove that if
$\mathcal{\mathcal{E}}$ has
Cartesian product structure in vector spaces over prime fields, then we can break the
exponent
$\left(d+1\right)\u22152$
and still cover all distances. We also show that the number of pairs of points in
$\mathcal{\mathcal{E}}$ of
any given distance is close to its expected value.
