有限循环群中集合的三重相关决定
On the determination of sets by their triple correlation in finite cyclic groups
摘要：
设$G$是一个有限阿贝尔群，$E$是它的一个子集，假设我们知道对于$G$的所有子集$T$，其大小直到$k$，对于有多少个$x \in G$，平移$x+T$包含在$E$中。此信息统称为$E$的$k$ -卡片组。人们可以自然地扩展$k$卡片组的定义域以包括$G$上的函数.给定组$G$，什么时候$G$中的集合的$k$ -deck足以确定转换的设置？2-deck是不够的（即使我们允许集合的反射，这也不会改变2-deck），第一个有趣的情况是$k=3$。我们进一步将$G$限制为循环的，并确定$n$的值，对于该值，$\ZZ_n$的子集的3层足以确定平移的设置。这就完成了Grünbaum和Moore开始的关于三层甲板的工作。我们还从上面估计了对于$\ZZ_n$的随机子集，存在另一个子集的概率，而不是第一个子集的平移，具有相同的3层。我们给一个指数小的上限时，以前已知的是$O(1\bigl / \sqrt{n})$。
英文摘要：
Let $G$ be a finite abelian group and $E$ a subset of it. Suppose that we know for all subsets $T$ of $G$ of size up to $k$ for how many $x \in G$ the translate $x+T$ is contained in $E$. This information is collectively called the $k$-deck of $E$. One can naturally extend the domain of definition of the $k$-deck to include functions on $G$. Given the group $G$ when is the $k$-deck of a set in $G$ sufficient to determine the set up to translation? The 2-deck is not sufficient (even when we allow for reflection of the set, which does not change the 2-deck) and the first interesting case is $k=3$. We further restrict $G$ to be cyclic and determine the values of $n$ for which the 3-deck of a subset of $\ZZ_n$ is sufficient to determine the set up to translation. This completes the work begun by Grünbaum and Moore as far as the 3-deck is concerned. We additionally estimate from above the probability that for a random subset of $\ZZ_n$ there exists another subset, not a translate of the first, with the same 3-deck. We give an exponentially small upper bound when the previously known one was $O(1\bigl / \sqrt{n})$.
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