Geodesic
A quantified version of Bourgain's sum-product estimate in \(\mathbb F_{p}\) for subsets of incomparable sizes
The electronic journal of combinatorics, Tome 15 (2008)
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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.
DOI : 10.37236/782
Classification : 11B75, 11T23 
@article{10_37236_782,
     author = {M. Z. Garaev},
     title = {A quantified version of {Bourgain's} sum-product estimate in \(\mathbb {F_{p}\)} for subsets of incomparable sizes},
     journal = {The electronic journal of combinatorics},
     year = {2008},
     volume = {15},
     doi = {10.37236/782},
     zbl = {1206.11031},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/782/}
}
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M. Z. Garaev. A quantified version of Bourgain's sum-product estimate in \(\mathbb F_{p}\) for subsets of incomparable sizes. The electronic journal of combinatorics, Tome 15 (2008). doi: 10.37236/782

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