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{\bf M. Z. Garaev}
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{\bf A Quantified Version of Bourgain's Sum-Product Estimate in ${\Bbb F}_p$ for Subsets of Incomparable Sizes}
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Let ${\Bbb F}_p$ be the field of residue classes modulo a prime
number $p.$  In this paper we prove that if $A,B\subset
{\Bbb F}_p^*,$ then for any fixed $\varepsilon>0,$
$$
|A+A|+|AB|\gg \Bigl(\min\Bigl\{|B|,\,
{p\over|A|}\Bigr\}\Bigr)^{1/25-\varepsilon}|A|.
$$
This quantifies Bourgain's recent sum-product estimate.



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