An improved sum-product estimate over finite fields

This paper gives an improved sum-product estimate for subsets of a finite field whose order is not prime. It is shown, under certain conditions, that $$\max \{\mathrm{\mid }A+A\mathrm{\mid },\mathrm{\mid }A\cdot A\mathrm{\mid }\}\gg \frac{\mathrm{\mid }A{\mathrm{\mid }}^{12\mathrm{/}11}}{({\log }_2\mathrm{\mid }A\mathrm{\mid }{)}^{5\mathrm{/}11}}\mathrm{.}$$ This new estimate matches, up to a logarithmic factor, the current best known bound obtained over prime fields by Rudnev (\cite{mishaSP}).