Geodesic
Quantitative sum product estimates on different sets
The electronic journal of combinatorics, Tome 15 (2008)
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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.
DOI : 10.37236/915
Classification : 11B75, 12E20
Mots-clés : sum-product estimate, sumset 
@article{10_37236_915,
     author = {Chun-Yen Shen},
     title = {Quantitative sum product estimates on different sets},
     journal = {The electronic journal of combinatorics},
     year = {2008},
     volume = {15},
     doi = {10.37236/915},
     zbl = {1171.11014},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/915/}
}
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Chun-Yen Shen. Quantitative sum product estimates on different sets. The electronic journal of combinatorics, Tome 15 (2008). doi: 10.37236/915

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