Generalizations of some zero sum theorems

Abstract

Given an abelian group G of order n, and a finite non-empty subset A of integers, the Davenport constant of G with weight A, denoted by D A(G), is defined to be the least positive integer t such that, for every sequence (x 1,.. ., x t) with x i ∈ G, there exists a non-empty subsequence (x j1,..., x jl) and a i ∈ A such that ∑ li=1 a i x ji = 0. Similarly, for an abelian group G of order n, E A(G) is defined to be the least positive integer t such that every sequence over G of length t contains a subsequence (x j1,..., x jn) such that ∑ ni=1 a i x ji = 0, for some a i ∈ A. When G is of order n, one considers A to be a non-empty subset of {1,..., n - 1}. If G is the cyclic group ℤ/nℤ, we denote E A(G) and D A(G) by E A(n) and D A(n) respectively. In this note, we extend some results of Adhikari et al (Integers 8 (2008) Article A52) and determine bounds for D Rn (n) and E Rn (n), where R n = {x 2: x ∈ (ℤ/nℤ)*}. We follow some lines of argument from Adhikari et al (Integers 8 (2008) Article A52) and use a recent result of Yuan and Zeng (European J. Combinatorics 31 (2010) 677-680), a theorem due to Chowla (Proc. Indian Acad. Sci. (Math. Sci.) 2 (1935) 242-243) and Kneser's theorem (Math. Z. 58 (1953) 459-484; 66 (1956) 88-110; 61 (1955) 429-434). © Indian Academy of Sciences.