Journal of the Ramanujan Mathematical Society

Volume 37, Issue 3, September 2022 pp. 281–286.

On some weighted zero-sum constants

Authors: Sukumar Das Adhikari and Shruti Hegde
Author institution:(Formerly at Harish-Chandra Research Institute), Department of Mathematics, Ramakrishna Mission Vivekananda Educational and Research Institute, Belur 711 202, India

Summary: Let G be a finite abelian group with exp(G) = n. For a positive integer k and a non-empty subset A of [1,n−1], the arithmetical invariant s{kn,A}(G) is defined to be the least positive integer t such that any sequence S of t elements in G has an A-weighted zero-sum subsequence of length kn. When A = {1} and k = 1, it is the Erdos-Ginzburg-Ziv constant and is denoted by s(G). The main result in this paper gives the exact value of s{kq,A}(G), for integers k ≥ 2 and A = {1,2}, where G is an abelian p-group with rank(G) ≤ 4, p is an odd prime and exp(G) = q. Our method consists of a modification of a polynomial method of Rónyai; by this method Rónyai had made a very good progress towards the Kemnitz conjecture (before Reiher proved the conjecture).

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