Geodesic
Sums of powers of subsets of an arbitrary finite field
Izvestiya. Mathematics , Tome 75 (2011) no. 2, pp. 253-285

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We discuss the following problem: given an integer $n\geqslant 2$, a real number $\varepsilon\in (0,1)$, and an arbitrary subset $A\subseteq\mathbb{F}_q$ which is not contained in a multiplicative shift of a proper subfield of $\mathbb{F}_q$ and satisfies $|A|>q^{\frac{1}{n-\varepsilon}}$, where $\mathbb{F}_q$ is the finite field of $q=p^r$ elements, describe those positive integers $N$ and $m$ for which we have a set-theoretic equality $NA^m=\mathbb{F}_q$. In particular, we show that this equality holds for $m=2n-2$ and $N=N(n,r,\varepsilon)$.
Keywords: sum-products of sets, finite field. 
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     author = {A. A. Glibichuk},
     title = {Sums of powers of subsets of an arbitrary finite field},
     journal = {Izvestiya. Mathematics },
     pages = {253--285},
     publisher = {mathdoc},
     volume = {75},
     number = {2},
     year = {2011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2011_75_2_a2/}
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A. A. Glibichuk. Sums of powers of subsets of an arbitrary finite field. Izvestiya. Mathematics , Tome 75 (2011) no. 2, pp. 253-285. http://geodesic.mathdoc.fr/item/IM2_2011_75_2_a2/