110 research outputs found
New sum-product type estimates over finite fields
Let be a field with positive odd characteristic . We prove a variety
of new sum-product type estimates over . They are derived from the theorem
that the number of incidences between points and planes in the
projective three-space , with , is where 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 such that
Comment: This is a revised version: Theorem 1 was incorrect as stated. We give
its correct statement; this does not seriously affect the main arguments
throughout the paper. Also added is a seres of remarks, placing the result in
the context of the current state of the ar
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