2023/10/31 by Krishnendu Sekhar Paul, Paul, Krishnendu, Shameek Paul +1
Computer Science · Mathematics · #11B50 #11B75 #Coding theory and cryptography #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematical Approximation and Integration #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2311.00090
openalex publication_date 2023/10/31 · openalex created_date 2023/11/03 · openalex updated_date 2026/07/28
Let A,B⊆\mathbb Zn be given and S=(x1,…, xk) be a sequence in \mathbb Zn. We say that S is an (A,B)-weighted zero-sum sequence if there exist a1,…,ak∈ A and b1,…,bk∈ B such that a1x1+⋯+akxk=0 and b1a1+⋯+bkak=0. We show that if S has length 2n-1, then S has an (A,B)-weighted zero-sum subsequence of length n. The constant EA,B is defined to be the smallest positive integer k such that every sequence of length k in \mathbb Zn has an (A,B)-weighted zero-sum subsequence of length n. A sequence in \mathbb Zn of length EA,B-1 which does not have any (A,B)-weighted zero-sum subsequence of length n is called an E-extremal sequence for (A,B). We determine the constant EA,B and characterize the E-extremal sequences for some pairs (A,B). We also study the related constants CA,B and DA,B which are defined in the article.