Geodesic
Properties of the sums and products of subsets in a finite field of prime order
Čebyševskij sbornik, Tome 8 (2007) no. 2, pp. 30-43
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It is proved that for any subsets $A_1,A_2,\ldots,A_n\subset\mathbb{F}_p, n\geqslant 2,$ such that $|A_i|\geqslant 2, 1\leqslant i\leqslant n,$ and $|A_1|\cdot |A_2|\cdot\ldots\cdot |A_n|>p^{1+\varepsilon}$ for some $\varepsilon>0$ we have $$NA_1\cdot A_2\cdot\ldots\cdot A_n=\mathbb{F}_p, $$ where $$N=\left\{ \begin{array}{ll} 16, \hbox{for $n=2$;} \\ 16\cdot\max\{1,24\left(\left[\log_2\left(\frac{1}{\varepsilon}\right)\right]+1\right)\}, \hbox{for $n=3$;} \\ 16^{n}\cdot\max\{7,2(-11-[\log_2(\varepsilon(n-2))])\}, \hbox{for $n>3$.} \\ \end{array} \right. $$ 
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     author = {A. A. Glibichuk},
     title = {Properties of the sums and products of subsets in a finite field of prime order},
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     url = {http://geodesic.mathdoc.fr/item/CHEB_2007_8_2_a3/}
}
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A. A. Glibichuk. Properties of the sums and products of subsets in a finite field of prime order. Čebyševskij sbornik, Tome 8 (2007) no. 2, pp. 30-43. http://geodesic.mathdoc.fr/item/CHEB_2007_8_2_a3/

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