关于素域扩张的一个注记
A note on expansion in prime fields
摘要：
设$β,ε\in (0,1]$和$k \geq \exp(122 \max\{1/β,1/ε\})$。证明了：若$A,B$是素域$\mathbb{Z}_{p}$和$|B| \geq p^β$的子集，则存在$$S = a_{1}B \pm \ldots \pm a_{k}B, \qquad a_{1},\ldots,a_{k} \in A,$$与$|S| \geq 2^{-12}p^{-ε}\min\{|A||B|,p\}$的和.作为推论，我们得到了下面的和积估计的初等证明。对于每个$α< 1$和$β,δ> 0$，存在$ε> 0$，使得以下成立。如果$A,B,E \subset \mathbb{Z}_{p}$满足$|A| \leq p^α$、$|B| \geq p^β$和$|B||E| \geq p^δ|A|$，则存在$t \in E$，使得对于某个绝对常数$c > 0$，有$$|A + tB| \geq c p^ε|A|,$$。一个更清晰的估计，多项式方法的基础上，从史蒂文斯和德Zeeuw最近的工作。
英文摘要：
Let $β,ε\in (0,1]$, and $k \geq \exp(122 \max\{1/β,1/ε\})$. We prove that if $A,B$ are subsets of a prime field $\mathbb{Z}_{p}$, and $|B| \geq p^β$, then there exists a sum of the form $$S = a_{1}B \pm \ldots \pm a_{k}B, \qquad a_{1},\ldots,a_{k} \in A,$$ with $|S| \geq 2^{-12}p^{-ε}\min\{|A||B|,p\}$. As a corollary, we obtain an elementary proof of the following sum-product estimate. For every $α< 1$ and $β,δ> 0$, there exists $ε> 0$ such that the following holds. If $A,B,E \subset \mathbb{Z}_{p}$ satisfy $|A| \leq p^α$, $|B| \geq p^β$, and $|B||E| \geq p^δ|A|$, then there exists $t \in E$ such that $$|A + tB| \geq c p^ε|A|,$$ for some absolute constant $c > 0$. A sharper estimate, based on the polynomial method, follows from recent work of Stevens and de Zeeuw.
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