On generalized Erdos-Ginzburg-Ziv constants of C-n(r)
On generalized Erdos-Ginzburg-Ziv constants of C-n(r)
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关于 C-n(r) 的广义 Erdos-Ginzburg-Ziv 常数
DOI：
10.1016/j.disc.2018.12.018
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发表时间：
2019
影响因子：
0.8
通讯作者：
Zhang Hanbin
中科院分区：
文献类型：
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作者：
Han Dongchun;Zhang Hanbin
设G是指数为exp（G）= n的可加有限阿贝尔群.对于任意正整数k，第k个Erdens-Ginzburg-Ziv常数skn（G）定义为使得G中长度至少为t的序列S都有长度为kn的零和子序列的最小正整数t.很容易看出s k n（C n r）≥（k+ r）n− r，其中n，r∈ N。Kubertin证明了这个等式对任何k≥ r都成立。本文证明了如下结果：·[（1）]对任意正整数k≥ 6，有skn（Cn 3）=（k+ 3）n+ O（nln n）. [(2)]对于任意的正整数k≥ 18，我们有skn（Cn 4）=（k+ 4）n+ O（nlnn）. [(3)]对n∈ N，设n的最大素幂因子为p a，其中a∈ N.对任意固定的r≥ 5，若对某个t∈ N，pt ≥ r，则对任意k∈ N，有s kptn（Cn r）≤（kpt + r）n+ crnlnn，其中cr是依赖于r的常数.在上述情形下，我们的结果渐近地验证了Kubertin的猜想.
Let G be an additive finite abelian group with exponent exp (G)= n. For any positive integer k, the k th Erdős–Ginzburg–Ziv constant s k n (G) is defined as the smallest positive integer t such that every sequence S in G of length at least t has a zero-sum subsequence of length k n. It is easy to see that s k n (C n r)≥(k+ r) n− r where n, r∈ N. Kubertin conjectured that the equality holds for any k≥ r. In this paper, we prove the following results:•[(1)] For every positive integer k≥ 6, we have s k n (C n 3)=(k+ 3) n+ O (n ln n).•[(2)] For every positive integer k≥ 18, we have s k n (C n 4)=(k+ 4) n+ O (n ln n).•[(3)] For n∈ N, assume that the largest prime power divisor of n is p a for some a∈ N. Forany fixed r≥ 5, if p t≥ r for some t∈ N, then for any k∈ N we have s k p t n (C n r)≤(k p t+ r) n+ c r n ln n, where c r is a constant that depends on r. Our results verify the conjecture of Kubertin asymptotically in the above cases.