关于循环群的完备子集
On complete subsets of the cyclic group
摘要：
阿贝尔$G$的子集$X$称为{ \em完备}，如果由$X$生成的子群的每个元素都可以表示为来自$X$的不同元素的非空和。设$A\subset \Z_n$使得$A$的所有元素与$n$互质。解决一个猜想的Erdens和海尔布隆，奥尔森证明，$A$是完整的，如果$n$是一个素数，如果$|A|>2\sqrt{n}.$最近武证明，有一个绝对常数$c$，这样，对于一个任意大的$n$，$A$是完整的，如果$|A|\ge c\sqrt{n},$，并重申，2基本上是正确的价值$c$。我们证明了$A$是完备的，如果$|A|> 1+2\sqrt{n-4}$，从而证明了最后一个猜想。
英文摘要：
A subset $X$ of an abelian $G$ is said to be {\em complete} if every element of the subgroup generated by $X$ can be expressed as a nonempty sum of distinct elements from $X$. Let $A\subset \Z_n$ be such that all the elements of $A$ are coprime with $n$. Solving a conjecture of Erdős and Heilbronn, Olson proved that $A$ is complete if $n$ is a prime and if $|A|>2\sqrt{n}.$ Recently Vu proved that there is an absolute constant $c$, such that for an arbitrary large $n$, $A$ is complete if $|A|\ge c\sqrt{n},$ and conjectured that 2 is essentially the right value of $c$. We show that $A$ is complete if $|A|> 1+2\sqrt{n-4}$, thus proving the last conjecture.
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