论文标题
$ \ mathbb z_n $的单位加权GAO常数的极端序列
Extremal sequences for the unit-weighted Gao constant of $\mathbb Z_n$
论文作者
论文摘要
对于$ a \ subseteq \ mathbb z_n $，$ a $ a $ weighted gao常数$ e_a（n）$被定义为最小的自然数量$ k $，从而使任何$ \ m mathbb z_n $中的任何$ k $ elements的序列$ k $ ementions in $ n $ n $ a $ n $均为$ a $ a $ a $ a-weater-wewtep sur in-weightew weite jurtectem s zere。长度$ e_a（n）-1 $ in $ \ mathbb z_n $的序列，没有任何$ a $ a $ a $ wate的零和长度的$ n $的零和子序列称为$ a $ a $ a $ a-extremal序列。据说具有$ n-1 $零的$ n-1 $零的序列是标准类型。当$ a = u（n）$（$ \ mathbb z_n $中的单位）$ n $是奇数时，我们表征所有这些序列并表明它们是标准类型。当$ n $甚至是$ n $时，我们给出了不标准类型的此类序列的示例。我们还表征了GAO常数的$ U（N）$ - 极端序列，当$ n = 2^rp $时，其中$ p $是一个奇怪的素数。
For $A\subseteq \mathbb Z_n$, the $A$-weighted Gao constant $E_A(n)$ is defined to be the smallest natural number $k$, such that any sequence of $k$ elements in $\mathbb Z_n$ has a subsequence of length $n$, whose $A$-weighted sum is zero. Sequences of length $E_A(n)-1$ in $\mathbb Z_n$, which do not have any $A$-weighted zero-sum subsequence of length $n$ are called $A$-extremal sequences for the Gao constant. Such a sequence which has $n-1$ zeroes is said to be of the standard type. When $A=U(n)$ (units in $\mathbb Z_n$) where $n$ is odd, we characterize all such sequences and show that they are of the standard type. When $n$ is even, we give examples of such sequences which are not of the standard type. We also characterize the $U(n)$-extremal sequences for the Gao constant, when $n=2^rp$, where $p$ is an odd prime.
