We prove that if $A \subset { \Bbb F}_q$ is such that $$|A|>q^{ { 1/2}+\frac{ 1}{ 2d}},$$ then $${ \Bbb F}_q^{ *} \subset dA^2=A^2+...+A^2 d \text{ times},$$ where $$A^2=\{ a \cdot a': a,a' \in A\},$$ and where ${ \Bbb F}_q^{ *}$ denotes the multiplicative group of the finite field ${ \Bbb F}_q$. In particular, we cover ${ \Bbb F}_q^{ *}$ by $A^2+A^2$ if $|A|>q^{ { 3/4}}$. Furthermore, we prove that if $$|A| \ge C_{ size}^{ \frac{ 1}{ d}}q^{ { 1/2}+\frac{ 1}{ 2(2d-1)}},$$ then $$|dA^2| \ge q \cdot \frac{ C^2_{ size}}{ C^2_{ size}+1}.$$ Thus $dA^2$ contains a positive proportion of the elements of ${ \Bbb F}_q$ under a considerably weaker size assumption.We use the geometry of ${ \Bbb F}_q^d$, averages over hyper-planes and orthogonality properties of character sums. In particular, we see that using operators that are smoothing on $L^2$ in the Euclidean setting leads to non-trivial arithmetic consequences in the context of finite fields.
Categories: math.NT math.CA