Some Zero Sum Problems in combinatorial number theory

Abstract

This thesis comprises of three results each of which dealt in separate chapters. First chapter is of introductory nature, as the title suggest. And the other three chapters are devoted to three different problems. Following is a brief introduction to our results. 1. Let G be any finite abelian group of rank r with invariants n1, n2, ? ? ? , nr. In other words, G = Zn1 _Zn2 _? ? ?_Znr where ni?s are integers satisfying 1 lt n1|n2| ? ? ? |nr. The Davenport constant of a group G is defined as the smallest positive integer t such that every sequence of length t of elements of G has a non-empty zero-sum subsequence. It has been conjectured by ´Sliwa that, D(G) _ Pr i=1 ni. Thinking in the direction of this conjecture we have obtained the following upper bound on Davenport constant D(G), of G, D(G) _ nr+nrand#8722;1+(c(3)and#8722;1)nrand#8722;2+(c(4)and#8722;1)nrand#8722;3+? ? ?+(c(r)and#8722;1)n1+1, where c(i)?s are Alon-Dubiner constants [10] for respective i?s. Also we shall give an application of Davenport?s constant to Quadratic sieve. 2. Let G be a finite abelian group with exp(G) = e. Let s(G) (respectively, _(G)) be the minimal positive integer t with the property that any sequence S of length t of elements of G contains an e-term subsequence (respectively, a non-empty subsequence of length at most e) of S with sum zero. For the group of rank at most two this constant has been determined completely (see [45]). Looking at the problem for groups of rank greater that 2 gave rise to this result. Our problem is to determine value of s(Cr nm) under some constraints on n,m, and r. Let n,m and r be positive integers and m _ 3. Furthermore, _(Cr m ) = ar(m and#8722; 1) + 1, for some constant ar depending on r and n is a fixed integer greater than or equal to, mr(c(r)m and#8722; ar(m and#8722; r) + m and#8722; 3)(m and#8722; 1) and#8722; (m + 1) + (m + 1)(ar + 1) m(m + 1)(ar + 1) and s(Cr n ) = (ar +1)(nand#8722;1)+1. In the above lower bound on n, c(r) is the Alon-Dubiner constant. Then s(Cr nm) = (ar + 1)(nm and#8722; 1) + 1. 3. Given an abelian group G of order n, and a finite non-empty subset A of integers, the Davenport...