无限群中的Erdos-Turan问题
The Erdos-Turan problem in infinite groups
摘要：
设$G$是一个无穷阿贝尔群，其上有$|2G|=|G|$ .证明了如果$G$不是指数为3的群与阶数为2的群的直和，则$G$具有完全可加基，即存在一个子集$S\subseteq G$，使得$G$的每个元素唯一地表示为$S$的两个元素之和.此外，如果$G$ \emph{is}是指数为3的群和阶数为2的群的直和，则它不具有完美的加法基;然而，在这种情况下，存在子集$S\subseteq G$，使得$G$的每个元素最多具有两个表示（在置换被加数的情况下是不同的）作为$S$的两个元素的和。这完全解决了无限群的Erdos-Turan问题。它还表明，如果$G$是指数为2的阿贝尔群，那么有一个子集$S\subseteq G$，使每个元素的$G$有一个表示为两个元素的和的$S$，和非零元素的表示的数量是有界的绝对常数。
英文摘要：
Let $G$ be an infinite abelian group with $|2G|=|G|$. We show that if $G$ is not the direct sum of a group of exponent 3 and the group of order 2, then $G$ possesses a perfect additive basis; that is, there is a subset $S\subseteq G$ such that every element of $G$ is uniquely representable as a sum of two elements of $S$. Moreover, if $G$ \emph{is} the direct sum of a group of exponent 3 and the group of order 2, then it does not have a perfect additive basis; however, in this case there is a subset $S\subseteq G$ such that every element of $G$ has at most two representations (distinct under permuting the summands) as a sum of two elements of $S$. This solves completely the Erdos-Turan problem for infinite groups. It is also shown that if $G$ is an abelian group of exponent 2, then there is a subset $S\subseteq G$ such that every element of $G$ has a representation as a sum of two elements of $S$, and the number of representations of non-zero elements is bounded by an absolute constant.
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