### A multiple set version of the 3k-3 theorem.

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A zero-sum sequence over ℤ is a sequence with terms in ℤ that sum to 0. It is called minimal if it does not contain a proper zero-sum subsequence. Consider a minimal zero-sum sequence over ℤ with positive terms $a\u2081,...,{a}_{h}$ and negative terms $b\u2081,...,{b}_{k}$. We prove that h ≤ ⌊σ⁺/k⌋ and k ≤ ⌊σ⁺/h⌋, where $\sigma \u207a={\sum}_{i=1}^{h}{a}_{i}=-{\sum}_{j=1}^{k}{b}_{j}$. These bounds are tight and improve upon previous results. We also show a natural partial order structure on the collection of all minimal zero-sum sequences over the set i∈ ℤ : -n ≤ i ≤ n for any positive integer n.

Let K be an algebraic number field with non-trivial class group G and ${}_{K}$ be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let ${F}_{k}\left(x\right)$ denote the number of non-zero principal ideals ${a}_{K}$ with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that ${F}_{k}\left(x\right)$ behaves, for x → ∞, asymptotically like $x{\left(logx\right)}^{1/\left|G\right|-1}{\left(loglogx\right)}^{{}_{k}\left(G\right)}$. In this article, it is proved that for every prime p, $\u2081\left({C}_{p}\oplus {C}_{p}\right)=2p$, and it is also proved that $\u2081\left({C}_{mp}\oplus {C}_{mp}\right)=2mp$ if $\u2081\left({C}_{m}\oplus {C}_{m}\right)=2m$ and m is large enough. In particular, it is shown that for...

We prove that every set A ⊂ ℤ satisfying ${\sum}_{x}min({1}_{A}*{1}_{A}\left(x\right),t)\le (2+\delta )t\left|A\right|$ for t and δ in suitable ranges must be very close to an arithmetic progression. We use this result to improve the estimates of Green and Morris for the probability that a random subset A ⊂ ℕ satisfies |ℕ∖(A+A)| ≥ k; specifically, we show that $\mathbb{P}\left(\right|\mathbb{N}\setminus (A+A)\left|\ge k\right)=\Theta \left({2}^{-k/2}\right)$.

We study the minimal number of elements of maximal order occurring in a zero-sumfree sequence over a finite Abelian p-group. For this purpose, and in the general context of finite Abelian groups, we introduce a new number, for which lower and upper bounds are proved in the case of finite Abelian p-groups. Among other consequences, our method implies that, if we denote by exp(G) the exponent of the finite Abelian p-group G considered, every zero-sumfree sequence S with maximal possible length over...

Suppose $A$ is a set of non-negative integers with upper Banach density $\alpha $ (see definition below) and the upper Banach density of $A+A$ is less than $2\alpha $. We characterize the structure of $A+A$ by showing the following: There is a positive integer $g$ and a set $W$, which is the union of $\lceil 2\alpha g-1\rceil $ arithmetic sequences [We call a set of the form $a+d\mathbb{N}$ an arithmetic sequence of difference $d$ and call a set of the form $\{a,a+d,a+2d,...,a+kd\}$ an arithmetic progression of difference $d$. So an arithmetic progression is finite and an arithmetic sequence...

We prove that there is a small but fixed positive integer $\u03f5$ such that for every prime $p$ larger than a fixed integer, every subset $S$ of the integers modulo $p$ which satisfies $\left|2S\right|\le (2+\u03f5)\left|S\right|$ and $2\left(\right|2S\left|\right)-2\left|S\right|+3\le p$ is contained in an arithmetic progression of length $\left|2S\right|-\left|S\right|+1$. This is the first result of this nature which places no unnecessary restrictions on the size of $S$.