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arXiv 0705.4256math.NTmath.CA
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有限域中的和与积：一个积分几何观点

Sums and products in finite fields: an integral geometric viewpoint

Derrick Hart, Alex Iosevich

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摘要：

证明了若$A \subset {\Bbb F}_q$使得$$|A|>q^{{1/2}+\frac{1}{2d}},$$则$${\Bbb F}_q^{*} \subset dA^2=A^2+...+A^2 d \text{times},$$，其中$$A^2=\{a \cdot a': a,a' \in A\},$$和${\Bbb F}_q^{*}$表示有限域${\Bbb F}_q$的乘法群.特别地，如果$|A|>q^{3/4}$，我们用$A^2+A^2$覆盖${\Bbb F}_q^{*}$。利用${\Bbb F}_q^d$的几何性质、超平面上的平均性质以及特征和的正交性质，证明了如果$$|A| \ge C_{size}^{\frac{1}{d}}q^{{1/2}+\frac{1}{2(2d-1)}},$$，则$$|dA^2| \ge q \cdot \frac{C^2_{size}}{C^2_{size}+1}.$$.特别是，我们看到，使用的运营商是在欧几里德设置$L^2$平滑导致非平凡的算术后果的上下文中的有限域。

英文摘要：

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.

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