New sum-product type estimates over finite fields

Let $F$ be a field with positive odd characteristic $p$. We prove a variety of new sum-product type estimates over $F$. They are derived from the theorem that the number of incidences between $m$ points and $n$ planes in the projective three-space $PG(3,F)$, with $m\geq n=O(p^2)$, is $$O( m\sqrt{n} + km ),$$ where $k$ denotes the maximum number of collinear planes. The main result is a significant improvement of the state-of-the-art sum-product inequality over fields with positive characteristic, namely that \begin{equation}\label{mres} |A\pm A|+|A\cdot A| =\Omega \left(|A|^{1+\frac{1}{5}}\right), \end{equation} for any $A$ such that $|A|<p^{\frac{5}{8}}.$