Properties of the sums and products of subsets in a finite field of prime order

A. A. Glibichuk

M. V. Lomonosov Moscow State University

Abstract: 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
$$N=\left\{
\begin{array}{ll} 16, & \hbox{for $n=2$;} \\ 16\cdot\max\{1,24\left(\left[\log_2\left(\frac{1}{\varepsilon}\right)\right]+1\right)\}, & \hbox{for $n=3$;} \\ 16^{n}\cdot\max\{7,2(-11-[\log_2(\varepsilon(n-2))])\}, & \hbox{for $n>3$.} \\ \end{array}
\right. $$


UDC: 511

MSC: 12E20

Received: 10.09.2007

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