arXiv math/06025682006-02-25math.COmath.NT

有限循环群中的长无零序列

Long zero-free sequences in finite cyclic groups

Svetoslav Savchev, Fang Chen

备注：: 13 pages

摘要：

一个可加可写阿贝尔群中的序列称为零自由的，如果它的每个非空序列的和不同于群的零元素。本文确定了模$n$的整数加法群$\Zn/$中长度大于$n/2$的无零序列的结构。主要结果表明，对于$\Zn/$中长度为$\ell>n/2$的每个无零序列$(a_i)_{i=1}^\ell$，存在与$n$互质的整数$g$，使得如果$\bar{ga_i}$表示同余类$ga_i$中的最小正整数（模$n$），则$Σ_{i=1}^\ell\bar{ga_i}<n$。对循环群的一些常见的零和问题的回答是直接的结果。在其他应用中，最好的可能的下界建立的最大多重性的一个零自由序列的长度大于$n/2$，以及为最大的多重性的生成器。该方法是组合的，并不呼吁以前已知的非平凡的事实。

英文摘要：

A sequence in an additively written abelian group is called zero-free if each of its nonempty subsequences has sum different from the zero element of the group. The article determines the structure of the zero-free sequences with lengths greater than $n/2$ in the additive group $\Zn/$ of integers modulo $n$. The main result states that for each zero-free sequence $(a_i)_{i=1}^\ell$ of length $\ell>n/2$ in $\Zn/$ there is an integer $g$ coprime to $n$ such that if $\bar{ga_i}$ denotes the least positive integer in the congruence class $ga_i$ (modulo $n$), then $Σ_{i=1}^\ell\bar{ga_i}<n$. The answers to a number of frequently asked zero-sum questions for cyclic groups follow as immediate consequences. Among other applications, best possible lower bounds are established for the maximum multiplicity of a term in a zero-free sequence with length greater than $n/2$, as well as for the maximum multiplicity of a generator. The approach is combinatorial and does not appeal to previously known nontrivial facts.

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