Sums and products in finite fields: an integral geometric viewpoint

Details Sums and products in finite fields: an integral geometric viewpoint Sums and products in finite fields: an integral geometric viewpoint

2007-05-29

arxiv.org/abs/0705.4256v4

Mathematics

Number Theory

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Sums and products in finite fields: an integral geometric viewpoint does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Sums and products in finite fields: an integral geometric viewpoint, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Sums and products in finite fields: an integral geometric viewpoint will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-499630