Generalizations of some zero sum theorems

作者：Chintamani M N^*; Moriya B K

来源：Proceedings of the Indian Academy of Sciences - Mathematical Sciences, 2012, 122(1): 15-21.

DOI：10.1007/s12044-012-0058-7

• Weighted zero sum problems
• Davenport constant
• EGZ theorem

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) is an element of G, there exists a non-empty subsequence (x(j1), ..., x(ji)) and a(i) is an element of A such that Sigma(l)(i=1)a(i)x(ji) = 0. Similarly, for an abelian group G of order n, E-A(G) is defined lobe the least positive integer t such that every sequence over G of length t contains a subsequence (x(j1), ..., x(jn)) such that Sigma(n)(i=1) a(i)x(ji) = 0, for some a(i) is an element of 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 eta! (Integers 8 (2008) Article A52) and determine bounds for D-Rn(n) and E-Rn(n), where R-n = {x(2) : x is an element of (Z/nZ)*}. We follow some lines of argument from Adhikari eta! (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).

• 出版日期2012-2