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
Voir la notice de l'article provenant de la source Math-Net.Ru
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
$$N=\left\{
\begin{array}{ll}
16, \hbox{for $n=2$;} \\
16\cdot\max\{1,24\left(\left[\log_2\left(\frac{1}{\varepsilon}\right)\right]+1\right)\}, \hbox{for $n=3$;} \\
16^{n}\cdot\max\{7,2(-11-[\log_2(\varepsilon(n-2))])\}, \hbox{for $n>3$.} \\
\end{array}
\right. $$
@article{CHEB_2007_8_2_a3,
author = {A. A. Glibichuk},
title = {Properties of the sums and products of subsets in a finite field of prime order},
journal = {\v{C}eby\v{s}evskij sbornik},
pages = {30--43},
publisher = {mathdoc},
volume = {8},
number = {2},
year = {2007},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/CHEB_2007_8_2_a3/}
}
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/