论文标题
关于Erd \ H OS，Kleitman和Lemke的反向问题
On an inverse problem of Erd\H os, Kleitman, and Lemke
论文作者
论文摘要
令$（g，1_g）$为有限组，让$ s = g_1 \ bdot \ ldots \ bdot \ bdot g _ {\ ell} $是$ g $的非空序列。我们说$ s $是一个很小的产品序列，如果可以订购其术语以使其产品等于$ 1_g $和$ \ sum_ {i = 1}^{\ ell} {\ ell} \ frac {1} {\ ord（g_i）} \ le 1 $。令$ \ mathsf {ti}（g）$为最小的整数$ t $，以使每个序列$ s $ the $ g $ a $ | s | s | \ ge t $具有一个很小的产品 - 一个子序列。直接问题是获得$ \ Mathsf {ti}（g）$的确切值，而逆问题是表征没有$ g $的长序列的结构，而$ g $没有微小的产品 - 一个子序列。在本文中，我们考虑了环状群体的逆问题，并且还研究了二面体和双环组的直接和反问题。
Let $(G, 1_G)$ be a finite group and let $S=g_1\bdot \ldots\bdot g_{\ell}$ be a nonempty sequence over $G$. We say $S$ is a tiny product-one sequence if its terms can be ordered such that their product equals $1_G$ and $\sum_{i=1}^{\ell}\frac{1}{\ord(g_i)}\le 1$. Let $\mathsf {ti}(G)$ be the smallest integer $t$ such that every sequence $S$ over $G$ with $|S|\ge t$ has a tiny product-one subsequence. The direct problem is to obtain the exact value of $\mathsf {ti}(G)$, while the inverse problem is to characterize the structure of long sequences over $G$ which have no tiny product-one subsequences. In this paper, we consider the inverse problem for cyclic groups and we also study both direct and inverse problems for dihedral groups and dicyclic groups.
