##### URI
http://hdl.handle.net/10355/5475
 dc.contributor.author Hart, Derrick, 1980- eng dc.contributor.author Iosevich, Alex, 1967- eng dc.date.issued 2007 eng dc.description http://arxiv.org/PS_cache/arxiv/pdf/0705/0705.4256v4.pdf eng dc.description.abstract 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. eng dc.identifier.citation arXiv:0705.4256v4 eng dc.identifier.uri http://hdl.handle.net/10355/5475 eng dc.language English eng dc.relation.ispartof Mathematics publications (MU) eng dc.relation.ispartofcommunity University of Missouri-Columbia. College of Arts and Sciences. Department of Mathematics eng dc.rights OpenAccess. eng dc.rights.license This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License. dc.subject classical analysis eng dc.subject.lcsh Number theory eng dc.title Sums and products in finite fields: an integral geometric viewpoint eng dc.type Article eng
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