Upper Bounds for the Davenport Constant

We prove that for all but a certain number of abelian groups of order n its Davenport constant is atmost n/k+k-1 for k=1,2,..,7. For groups of order three we improve on the existing bound involving the Alon-Dubiner constant.Comment: article soumis, decembre 200

Some Results on Zero Sum Sequences in Zp3Z_p^3

Kemnitz Conjecture [9] states that if we take a sequence of elements in $Z_{p}^{2}$ of length $4p-3$, $p$ is a prime number, then it has a subsequence of length $p$, whose sum is $0$ modulo $p$. It is known that in $Z_{p}^{3}$ to get a similar result we have to take a sequence of length atleast $9p-8$ . In this paper we will show that if we add a condition on the chosen sequence, then we can get a good upper and a lower bound for which similar results hold.Comment: This was a part of my master thesis under Prof. Gautami Bhowmi

On the zero-sum constant, the Davenport constant and their analogues

Let $D(G)$ be the Davenport constant of a finite Abelian group $G$. For a positive integer $m$ (the case $m = 1$, is the classical one) let ${\mathsf E}_m(G)$ (or $\eta_m(G)$, respectively) be the least positive integer $t$ such that every sequence of length $t$ in $G$ contains $m$ disjoint zero-sum sequences, each of length $|G|$ (or of length $\le exp(G)$ respectively). In this paper, we prove that if $G$ is an~Abelian group, then ${\mathsf E}_m(G)=D(G)-1+m|G|$, which generalizes Gao's relation. We investigate also the non-Abelian case. Moreover, we examine the asymptotic behavior of the sequences $({\mathsf E}_m(G))_{m\ge 1}$ and $(\eta_m(G))_{m\ge 1}.$ We prove a~generalization of Kemnitz's conjecture. The paper also contains a result of independent interest, which is a stronger version of a result by Ch. Delorme, O. Ordaz, D. Quiroz. At the and we apply the Davenport constant to smooth numbers and make a natural conjecture in the non-Abelian case.Comment: 16 page

On a combinatorial problem of Erdos, Kleitman and Lemke

In this paper, we study a combinatorial problem originating in the following conjecture of Erdos and Lemke: given any sequence of n divisors of n, repetitions being allowed, there exists a subsequence the elements of which are summing to n. This conjecture was proved by Kleitman and Lemke, who then extended the original question to a problem on a zero-sum invariant in the framework of finite Abelian groups. Building among others on earlier works by Alon and Dubiner and by the author, our main theorem gives a new upper bound for this invariant in the general case, and provides its right order of magnitude.Comment: 15 page

The structure of maximal zero-sum free Sequences

Let n be an integer, and consider finite sequences of elements of the group Z/nZ x Z/nZ. Such a sequence is called zero-sum free, if no subsequence has sum zero. It is known that the maximal length of such a zero-sum free sequence is 2n-2, and Gao and Geroldinger conjectured that every zero-sum free sequence of this length contains an element with multiplicity at least n-2. By recent results of Gao, Geroldinger and Grynkiewicz, it essentially suffices to verify the conjecture for n prime. Now fix a sequence (a_i) of length 2n-2 with maximal multiplicity of elements at most n-3. There are different approeaches to show that (a_i) contains a zero-sum; some work well when (a_i) does contain elements with high multiplicity, others work well when all multiplicities are small. The aim of this article is to initiate a systematic approach to property B via the highest occurring multiplicities. Our main results are the following: denote by m_1 >= m_2 the two maximal multiplicities of (a_i), and suppose that n is sufficiently big and prime. Then (a_i) contains a zero-sum in any of the following cases: when m_2 >= 2/3n, when m_1 > (1-c)n, and when m_2 < cn, for some constant c > 0 not depending on anything.Comment: 27 pages, 3 figure