Journal of the Ramanujan Mathematical Society

Volume 39, Issue 4, December 2024 pp. 369–376.

Generalization of some weighted zero-sum theorems and related extremal sequence

Authors: Subha Sarkar
Author institution:Department of Mathematics, National Institute of Technology Jamshedpur, Jharkhand 831 014, India.

Summary: Let G be a finite abelian group of exponent n and let A be a non-empty subset of [1, n−1]. The Davenport constant of G with weight A, denoted by D{A}(G), is defined to be the least positive integer 𝓁 such that any sequence over G of length 𝓁 has a non-empty A-weighted zero-sum subsequence. Similarly, the combinatorial constant E{A}(G) is defined to be the least positive integer 𝓁 such that any sequence over G of length 𝓁 has an A-weighted zero-sum subsequence of length |G|. In this article, we determine the exact value of D{A}(ℤ{n}), for some values of n, where A is the set of all cubes in ℤ{n}{∗}. We also determine the structure of the related extremal sequence in this case.

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