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 |