论文标题
小小的扭转组的小组加倍
Small doubling in groups with moderate torsion
论文作者
论文摘要
我们确定了一个有限的子集$ a $ a $ a abelian组的结构，因为$ | 2a | <3（1-ε）| a | $，$ε> 0 $;也就是说，我们表明$ a $包含在“小”一维坐标进程中，或者在有限亚组的$ε^{ - 1} $ cosets的结合中包含。 边界$ 3（1-ε）| a | $和$ε^{ - 1} $是最好的，因为没有一个不加紧另一个，并且对包含$ a $的coset进程大小的估计值很清晰。 在基础群体是无限循环的情况下，我们的结果将减少到著名的弗莱曼（Freiman）的$（3N-3）$ - 定理；因此，前者可以被视为将后者扩展到任意的阿贝尔群体上，前提是“不涉及太多扭转”。
We determine the structure of a finite subset $A$ of an abelian group given that $|2A|<3(1-ε)|A|$, $ε>0$; namely, we show that $A$ is contained either in a "small" one-dimensional coset progression, or in a union of fewer than $ε^{-1}$ cosets of a finite subgroup. The bounds $3(1-ε)|A|$ and $ε^{-1}$ are best possible in the sense that none of them can be relaxed without tightened another one, and the estimate obtained for the size of the coset progression containing $A$ is sharp. In the case where the underlying group is infinite cyclic, our result reduces to the well-known Freiman's $(3n-3)$-theorem; the former thus can be considered as an extension of the latter onto arbitrary abelian groups, provided that there is "not too much torsion involved".
