2010/03/10 by Sukumar Das Adhikari, Adhikari, Sukumar Das, David J. Grynkiewicz +3 · 1 citation
Engineering · Mathematics · #05D05 #11B75 #20D60 #20K01 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Limits and Structures in Graph Theory #Number Theory (math.NT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1003.2186
openalex publication_date 2010/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a finite additive abelian group with exponent exp(G)=n>1 and let A be a nonempty subset of 1,...,n-1. In this paper, we investigate the smallest positive integer m, denoted by sA(G), such that any sequence cii=1m with terms from G has a length n=exp(G) subsequence cijj=1n for which there are a1,...,an in A such that sumj=1naicij=0. When G is a p-group, A contains no multiples of p and any two distinct elements of A are incongruent mod p, we show that sA(G) is at most \lceil D(G)/|A|\rceil+exp(G)-1 if |A| is at least (D(G)-1)/(exp(G)-1), where D(G) is the Davenport constant of G and this upper bound for sA(G)in terms of |A| is essentially best possible. In the case A=1,-1, we determine the asymptotic behavior of s1,-1(G) when exp(G) is even, showing that, for finite abelian groups of even exponent and fixed rank, s1,-1(G)=exp(G)+log2|G|+O(log2log2|G|) as exp(G) tends to the infinity. Combined with a lower bound of exp(G)+sumi=1r\lfloorlog2 ni\rfloor, where G=\Zn1⊕...⊕ \Znr with 1