Related papers: On the number of fully weighted zero-sum subsequen…

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,…

Let $G$ be a group. The subsets $A_1,\ldots,A_k$ of $G$ form a complete factorization of group $G$ if if they are pairwise disjoint and each element $g\in G$ is uniquely represented as $g=a_1\ldots a_k$, with $a_i\in A_i$. We prove the…

Let $G$ be a finite cyclic group. Every sequence $S$ of length $l$ over $G$ can be written in the form $S=(n_1g)\cdot\ldots\cdot(n_lg)$ where $g\in G$ and $n_1, \ldots, n_l\in[1, \ord(g)]$, and the index $\ind(S)$ of $S$ is defined to be…

A group $G$ is said to be factorized into subsets $A_1, A_2, \ldots, A_s\subseteq G$ if every element $g$ in $G$ can be uniquely represented as $g=g_1g_2\ldots g_s$, where $g_i\in A_i$, $i=1,2,\ldots,s$. We consider the following…

Let $A_1,\ldots,A_n$ be finite subsets of an additive abelian group $G$ with $|A_1|=\cdots=|A_n|\ge2$. Concerning the two new kinds of restricted sumsets $$L(A_1,\ldots,A_n)=\{a_1+\cdots+a_n:\ a_1\in A_1,\ldots,a_n\in A_n,\ \text{and}\…

Let $G$ be an abelian group, and let $\mathcal F (G)$ be the free commutative monoid with basis $G$. For $\Omega \subset \mathcal F (G)$, define the universal zero-sum invariant ${\mathsf d}_{\Omega}(G)$ to be the smallest integer $\ell$…

Let $G$ be a finite abelian group, and let $m>0$ with $\exp(G)\mid m$. Let $s_{m}(G)$ be the generalized Erd\H{o}s-Ginzburg-Ziv invariant which denotes the smallest positive integer $d$ such that any sequence of elements in $G$ of length…

Let $G$ be a finite abelian group and $s$ be a positive integer. A subset $A$ of $G$ is called a {\em perfect $s$-basis of $G$} if each element of $G$ can be written uniquely as the sum of at most $s$ (not-necessarily-distinct) elements of…

Let $\mathcal S$ be a multiset of integers. We say $\mathcal S$ is a $\textit{zero-sum sequence}$ if the sum of its elements is 0. We study zero-sum sequences whose elements lie in the interval $[-k,k]$ such that no subsequence of length…

For an additive group $\Gamma$ the sequence $S = (g_1, \ldots, g_t)$ of elements of $\Gamma$ is a zero-sum sequence if $g_1 + \cdots + g_t = 0_\Gamma$. The cross number of $S$ is defined to be the sum $\sum_{i=1}^k 1/|g_i|$, where $|g_i|$…

Let $G$ be a finite cyclic group. Every sequence $S$ of length $l$ over $G$ can be written in the form $S=(x_1g)\cdot\ldots\cdot(x_lg)$ where $g\in G$ and $x_1, \ldots, x_l\in[1, \ord(g)]$, and the index $\ind(S)$ of $S$ is defined to be…

Let $G$ be an additive abelian group and $S\subset G$ a subset. Let $\Sigma(S)$ denote the set of group elements which can be expressed as a sum of a nonempty subset of $S$. We say $S$ is zero-sum free if $0 \not\in \Sigma(S)$. It was…

Let $t$ and $k$ be a positive integers, and let $I_k=\{i\in \mathbb{Z}:\; -k\leq i\leq k\}$. Let $\mathsf{s}'_t(I_k)$ be the smallest positive integer $\ell$ such that every zero-sum sequence $S$ over $I_k$ of length $|S|\ge \ell$ contains…

Let $t\geq 1$, let $A$ and $B$ be finite, nonempty subsets of an abelian group $G$, and let $A\pp{i} B$ denote all the elements $c$ with at least $i$ representations of the form $c=a+b$, with $a\in A$ and $b\in B$. For $|A|, |B|\geq t$, we…

Let $G$ be a finite abelian group with $\exp(G)$ the exponent of $G$. Then $\mathsf W(G)$ denotes the set of cross numbers of minimal zero-sum sequences over $G$ and $\mathsf w(G)$ denotes the set of all cross numbers of non-trivial…

Denoting by Sigma(S) the set of subset sums of a subset S of a finite abelian group G, we prove that |Sigma(S)| >= |S|(|S|+2)/4-1 whenever S is symmetric, |G| is odd and Sigma(S) is aperiodic. Up to an additive constant of 2 this result is…

Given a finite abelian group $G$ and a subset $S\subseteq G$, we let $N_{G,\ S}$ be the smallest integer $N$ such that for any subset $A\subseteq G$ with $N$ elements, we have $g+S\subseteq A$ for some $g\in G$. Using the probabilistic…

Let $\mathcal C$ be a set of finite groups which is closed under taking subgroups and let $d$ and $M$ be positive integers. Suppose that for any $G\in\mathcal C$ whose order is divisible by at most two distinct primes there exists an…

It is well-known that any sequence of at least N integers contains a subsequence whose sum is 0 (mod N). However, there can be very few subsequences with this property (e.g. if the initial sequence is just N 1's, then there is only one…

We study several combinatorial properties of finite groups that are related to the notions of sequenceability, R-sequenceability, and harmonious sequences. In particular, we show that in every abelian group $G$ with a unique involution…