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 $(x_{j_1},\ldots, x_{j_l})$ and a _i  ∈ A such that $\sum_{i=1}^{l}a_ix_{j_i} = 0$. 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 $(x_{j_1} ,\ldots, x_{j_n})$ such that $\sum_{i=1}^{n}a_ix_{j_i} = 0$, 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 ${\Bbb Z}/n{\Bbb Z}$, 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 $D_{R_n}(n)$ and $E_{R_n}(n)$, where $R_n = \{x^2 : x \in (\mathbb{Z}/n\mathbb{ Z})^*\}$. 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).