Abstract Let S = g 1 ⋅ … ⋅ g n S={g}_{1}\cdot \ldots \cdot {g}_{n} be a sequence with elements g i {g}_{i} from an additive finite abelian group G. S is called a tiny zero-sum sequence if S is non-empty, g 1 + … + g n = 0 {g}_{1}+\hspace{0.2em}\ldots \hspace{0.2em}+{g}_{n}=0 and k ( S ) ≔ ∑ i = 1 n 1 ord ( g i ) ≤ 1 k(S):= {\sum }_{i=1}^{n}\frac{1}{\text{ord}({g}_{i})}\le 1 . Let t ( G ) t(G) be the smallest integer t such that every sequence of t elements (repetition allowed) from G contains a tiny zero-sum sequence. In this article, we mainly focus on the explicit value of t ( G ) t(G) and compute this value for a new class of groups, namely ones of the form G = C 3 ⊕ C 3 p G={C}_{3}\oplus {C}_{3p} , where p is a prime number such that p ≥ 5 p\ge 5 .

中文翻译：

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一些特殊群上的微小零和序列

摘要 令 S = g 1 ⋅ … ⋅ gn S={g}_{1}\cdot \ldots \cdot {g}_{n} 是一个序列，其元素为 gi {g}_{i} 来自加法有限阿贝尔群 G。如果 S 是非空的，则 S 称为微小零和序列，g 1 + … + gn = 0 {g}_{1}+\hspace{0.2em}\ldots \hspace{0.2em}+ {g}_{n}=0 且 k ( S ) ≔ ∑ i = 1 n 1 ord ( gi ) ≤ 1 k(S):= {\sum }_{i=1}^{n}\frac{ 1}{\text{ord}({g}_{i})}\le 1 . 令 t ( G ) t(G) 是最小的整数 t，使得来自 G 的每个 t 元素序列（允许重复）包含一个微小的零和序列。在本文中，我们主要关注 t ( G ) t(G) 的显式值，并为一类新的组计算该值，即 G = C 3 ⊕ C 3 p G={C}_ {3}\oplus {C}_{3p} ，其中 p 是一个质数，使得 p ≥ 5 p\ge 5 。