Sum-product estimates in finite fields
  We prove, using combinatorics and Kloosterman sum technology that if $A \subset {\Bbb F}_q$, a finite field with $q$ elements, and $q^{{1/2}} \lesssim |A| \lesssim q^{{7/10}}$, then $\max \{|A+A|, |A \cdot A|\} \gtrsim \frac{{|A|}^{{3/2}}}{q^{{1/4}}$. 
D. Hart, A. Iosevich and J. Solymosi