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arXiv 1702.02544math.DSmath.COmath.NT
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关于正密度集的差集积

On product of difference sets for sets of positive density

Alexander Fish

备注：
6 pages; a new proof (due to I. Shkredov) of Lemma 2.1 that does not use Szemeredi's theorem has been added
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摘要：

本文证明了给定两个正密度集$E_1,E_2 \subset \mathbb{Z}$，存在$k \geq 1$，它被一个仅依赖于$E_1$和$E_2$的密度的数所包围，使得$k\mathbb{Z} \subset (E_1-E_1)\cdot(E_2-E_2)$ .作为主要定理的推论，我们推出，如果$α,β> 0$，那么存在$N_0$和$d_0$，它们只依赖于$α$和$β$，使得对于每个具有$|E_1| \geq αN, |E_2| \geq βN$的$N \geq N_0$和$E_1,E_2 \subset \mathbb{Z}_N$，存在$d \leq d_0$，$N$的因子满足$d \, \mathbb{Z}_N \subset (E_1-E_1)\cdot(E_2-E_2)$。

英文摘要：

In this paper we prove that given two sets $E_1,E_2 \subset \mathbb{Z}$ of positive density, there exists $k \geq 1$ which is bounded by a number depending only on the densities of $E_1$ and $E_2$ such that $k\mathbb{Z} \subset (E_1-E_1)\cdot(E_2-E_2)$. As a corollary of the main theorem we deduce that if $α,β> 0$ then there exist $N_0$ and $d_0$ which depend only on $α$ and $β$ such that for every $N \geq N_0$ and $E_1,E_2 \subset \mathbb{Z}_N$ with $|E_1| \geq αN, |E_2| \geq βN$ there exists $d \leq d_0$ a divisor of $N$ satisfying $d \, \mathbb{Z}_N \subset (E_1-E_1)\cdot(E_2-E_2)$.

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