关于移位迭代积集II
On iterated product sets with shifts II
摘要：
本文的主要结果是：对所有的$b \in \mathbb Z$，存在$k=k(b)$，使得对任意有限的$A \subset \mathbb Q$和任意非零的$u \in \mathbb Q$，有\[ \max \{ |A^{(k)}|, |(A+u)^{(k)}| \} \geq |A|^b, \].这里，$|A^{(k)}|$表示$k$ -折叠乘积集$\{a_1\cdots a_k : a_1, \dots, a_k \in A \}$ .此外，我们的证明方法还给出了以下$l_{\infty}$和积估计。对于所有的$γ>0$，存在一个常数$C=C(γ)$，使得对于任何具有$|AA| \leq K|A|$和任何$c_1,c_2 \in \mathbb Q \setminus \{0\}$的$A \subset \mathbb Q$，最多有\[ c_1x + c_2y =1 ,\,\,\,\,\,\,\, (x,y) \in A \times A. \]的$K^C|A|^γ$解。特别地，当$K=|A|^ε$时，这个结果给出了一个强界，只要$ε>0$足够小，从而改进了以前通过子空间定理获得的界。在进一步的应用中，我们给出了一个部分结构定理的点集，确定许多发病率，并证明了和集增长任意大，采取足够多的产品。我们利用了Zhelezov和Pálvölgyi提出的多项式Freiman-Ruzsa猜想的查询复杂性模拟。这个新工具取代了我们以前使用的Bourgain和Chang的复杂设置。此外，参数之间存在更好的定量依赖性。
英文摘要：
The main result of this paper is the following: for all $b \in \mathbb Z$ there exists $k=k(b)$ such that \[ \max \{ |A^{(k)}|, |(A+u)^{(k)}| \} \geq |A|^b, \] for any finite $A \subset \mathbb Q$ and any non-zero $u \in \mathbb Q$. Here, $|A^{(k)}|$ denotes the $k$-fold product set $\{a_1\cdots a_k : a_1, \dots, a_k \in A \}$. Furthermore, our method of proof also gives the following $l_{\infty}$ sum-product estimate. For all $γ>0$ there exists a constant $C=C(γ)$ such that for any $A \subset \mathbb Q$ with $|AA| \leq K|A|$ and any $c_1,c_2 \in \mathbb Q \setminus \{0\}$, there are at most $K^C|A|^γ$ solutions to \[ c_1x + c_2y =1 ,\,\,\,\,\,\,\, (x,y) \in A \times A. \] In particular, this result gives a strong bound when $K=|A|^ε$, provided that $ε>0$ is sufficiently small, and thus improves on previous bounds obtained via the Subspace Theorem. In further applications we give a partial structure theorem for point sets which determine many incidences and prove that sum sets grow arbitrarily large by taking sufficiently many products. We utilise a query-complexity analogue of the polynomial Freiman-Ruzsa conjecture, due to Zhelezov and Pálvölgyi. This new tool replaces the role of the complicated setup of Bourgain and Chang, which we had previously used. Furthermore, there is a better quantitative dependence between the parameters.
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