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

Some remarks on barycentric-sum problems over cyclic groups

We derive some new results on the k-th barycentric Olson constants of abelian groups (mainly cyclic). This quantity, for a finite abelian (additive) group (G,+), is defined as the smallest integer l such that each subset A of G with at least l elements contains a subset with k elements {g_1, ..., g_k} satisfying g_1 + ... + g_k = k g_j for some 1 <= j <= k.Comment: to appear in European Journal of Combinatoric

Representation of Finite Abelian Group Elements by Subsequence Sums

Let $G\cong C_{n_1}\oplus ... \oplus C_{n_r}$ be a finite and nontrivial abelian group with $n_1|n_2|...|n_r$. A conjecture of Hamidoune says that if $W=w_1... w_n$ is a sequence of integers, all but at most one relatively prime to $|G|$, and $S$ is a sequence over $G$ with $|S|\geq |W|+|G|-1\geq |G|+1$, the maximum multiplicity of $S$ at most $|W|$, and $\sigma(W)\equiv 0\mod |G|$, then there exists a nontrivial subgroup $H$ such that every element $g\in H$ can be represented as a weighted subsequence sum of the form $g=\sum_{i=1}^{n}w_is_i$, with $s_1... s_n$ a subsequence of $S$. We give two examples showing this does not hold in general, and characterize the counterexamples for large $|W|\geq {1/2}|G|$. A theorem of Gao, generalizing an older result of Olson, says that if $G$ is a finite abelian group, and $S$ is a sequence over $G$ with $|S|\geq |G|+D(G)-1$, then either every element of $G$ can be represented as a $|G|$-term subsequence sum from $S$, or there exists a coset $g+H$ such that all but at most $|G/H|-2$ terms of $S$ are from $g+H$. We establish some very special cases in a weighted analog of this theorem conjectured by Ordaz and Quiroz, and some partial conclusions in the remaining cases, which imply a recent result of Ordaz and Quiroz. This is done, in part, by extending a weighted setpartition theorem of Grynkiewicz, which we then use to also improve the previously mentioned result of Gao by showing that the hypothesis $|S|\geq |G|+D(G)-1$ can be relaxed to $|S|\geq |G|+d^*(G)$, where d^*(G)=\Sum_{i=1}^{r}(n_i-1). We also use this method to derive a variation on Hamidoune's conjecture valid when at least $d^*(G)$ of the $w_i$ are relatively prime to $|G|$

Arithmetic-Progression-Weighted Subsequence Sums

Let $G$ be an abelian group, let $S$ be a sequence of terms $s_1,s_2,...,s_{n}\in G$ not all contained in a coset of a proper subgroup of $G$, and let $W$ be a sequence of $n$ consecutive integers. Let $$W\odot S=\{w_1s_1+...+w_ns_n:\;w_i {a term of} W,\, w_i\neq w_j{for} i\neq j\},$$ which is a particular kind of weighted restricted sumset. We show that $|W\odot S|\geq \min\{|G|-1,\,n\}$, that $W\odot S=G$ if $n\geq |G|+1$, and also characterize all sequences $S$ of length $|G|$ with $W\odot S\neq G$. This result then allows us to characterize when a linear equation $$a_1x_1+...+a_rx_r\equiv \alpha\mod n,$$ where $\alpha,a_1,..., a_r\in \Z$ are given, has a solution $(x_1,...,x_r)\in \Z^r$ modulo $n$ with all $x_i$ distinct modulo $n$. As a second simple corollary, we also show that there are maximal length minimal zero-sum sequences over a rank 2 finite abelian group $G\cong C_{n_1}\oplus C_{n_2}$ (where $n_1\mid n_2$ and $n_2\geq 3$) having $k$ distinct terms, for any $k\in [3,\min\{n_1+1,\,\exp(G)\}]$. Indeed, apart from a few simple restrictions, any pattern of multiplicities is realizable for such a maximal length minimal zero-sum sequence