On generalized Erdős-Ginzburg-Ziv constants of C_n^r

2018/09/18 by Dongchun Han, Han, Dongchun, Hanbin Zhang +1
Engineering · Mathematics · #11B75 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #Number Theory (math.NT) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1809.06548

openalex publication_date 2018/09/18 · openalex created_date 2018/09/27 · openalex updated_date 2026/07/28

Abstract

Let G be an additive finite abelian group with exponent exp(G)=m. For any positive integer k, the k-th generalized Erdős-Ginzburg-Ziv constant \mathsf s_km(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 km. It is easy to see that \mathsf s_kn(C_n^r)≥(k+r)n-r where n,r∈\mathbb N. Kubertin conjectured that the equality holds for any k≥ r. In this paper, we mainly prove the following results: (1) For every positive integer k≥ 6, we have \mathsf s_kn(C_n³)=(k+3)n+O((n)/(ln n)). (2) For every positive integer k≥ 18, we have \mathsf s_kn(C_n⁴)=(k+4)n+O((n)/(ln n)). (3) For n∈ \mathbb N, assume that the largest prime power divisor of n is p^a for some a∈\mathbb N. For any fixed r≥ 5, if p^t≥ r for some t∈\mathbb N, then for any k∈\mathbb N we have \mathsf s_kp^tn(C_n^r)≤(kp^t+r)n+c_r(n)/(ln n), where c_r is a constant depends on r. Note that the main terms in our results are consistent with the conjectural values proposed by Kubertin.

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