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

Zero-sum problems for abelian p-groups and covers of the integers by residue classes

Zero-sum problems for abelian groups and covers of the integers by residue classes, are two different active topics initiated by P. Erdos more than 40 years ago and investigated by many researchers separately since then. In an earlier announcement [Electron. Res. Announc. Amer. Math. Soc. 9(2003), 51-60], the author claimed some surprising connections among these seemingly unrelated fascinating areas. In this paper we establish further connections between zero-sum problems for abelian p-groups and covers of the integers. For example, we extend the famous Erdos-Ginzburg-Ziv theorem in the following way: If {a_s(mod n_s)}_{s=1}^k covers each integer either exactly 2q-1 times or exactly 2q times where q is a prime power, then for any c_1,...,c_k in Z/qZ there exists a subset I of {1,...,k} such that sum_{s in I}1/n_s=q and sum_{s in I}c_s=0. Our main theorem in this paper unifies many results in the two realms and also implies an extension of the Alon-Friedland-Kalai result on regular subgraphs

Kemnitz’ conjecture revisited

AbstractA conjecture of Kemnitz remained open for some 20 years: each sequence of 4n-3 lattice points in the plane has a subsequence of length n whose centroid is a lattice point. It was solved independently by Reiher and di Fiore in the autumn of 2003. A refined and more general version of Kemnitz’ conjecture is proved in this note. The main result is about sequences of lengths between 3p-2 and 4p-3 in the additive group of integer pairs modulo p, for the essential case of an odd prime p. We derive structural information related to their zero sums, implying a variant of the original conjecture for each of the lengths mentioned. The approach is combinatorial

Zero-sum problems with congruence conditions

For a finite abelian group $G$ and a positive integer $d$, let $\mathsf s_{d \mathbb N} (G)$ denote the smallest integer $\ell \in \mathbb N_0$ such that every sequence $S$ over $G$ of length $|S| \ge \ell$ has a nonempty zero-sum subsequence $T$ of length $|T| \equiv 0 \mod d$. We determine $\mathsf s_{d \mathbb N} (G)$ for all $d\geq 1$ when $G$ has rank at most two and, under mild conditions on $d$, also obtain precise values in the case of $p$-groups. In the same spirit, we obtain new upper bounds for the Erd{\H o}s--Ginzburg--Ziv constant provided that, for the $p$-subgroups $G_p$ of $G$, the Davenport constant $\mathsf D (G_p)$ is bounded above by $2 \exp (G_p)-1$. This generalizes former results for groups of rank two