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{\bf Chun-Yen Shen}
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{\bf Quantitative Sum Product Estimates on Different Sets}
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Let $F_p$ be a finite field of $p$ elements with $p$ prime. In this
paper we show that for $A ,B \subset F_p$ with $|B|\leq |A| <
p^{{1 \over 2}}$ then
$$\max\big(|A+B|, |AB|\big) \gtrapprox 
\bigg({|B|^{14} \over |A|^{13}}\bigg)^{1/18}|A|.$$
This gives an explicit exponent in a sum-product estimate for
different sets by Bourgain.



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