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1
Content available remote

On A Combinatorial Problem Connected withFactorizations

100%
Gao W.
Colloquium Mathematicum
|
1997
|
tom 72
|
nr 2
251-268
2
Content available remote

Inverse zero-sum problems {III}

51%
Gao W., Geroldinger A., Grynkiewicz D.
Acta Arithmetica
|
2010
|
tom 141
|
nr 2
103-152
3
Content available remote

Inverse zero-sum problems

51%
Gao W., Geroldinger A., Schmid W.
Acta Arithmetica
|
2007
|
tom 128
|
nr 3
245-279
4
Content available remote

A quantitative aspect of non-unique factorizations: the Narkiewicz constants II

51%
Gao W., Li Y., Peng J.
Colloquium Mathematicum
|
2011
|
tom 124
|
nr 2
205-218
EN
Let K be an algebraic number field with non-trivial class group G and $𝓞_{K}$ be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let $F_{k}(x)$ denote the number of non-zero principal ideals $a𝓞_{K}$ with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that $F_{k}(x)$ behaves, for x → ∞, asymptotically like $x(log x)^{1/|G|-1} (loglogx)^{𝖭_{k}(G)}$. In this article, it is proved that for every prime p, $𝖭₁(C_{p}⊕ C_{p}) = 2p$, and it is also proved that $𝖭₁ (C_{mp}⊕ C_{mp}) = 2mp$ if $𝖭₁ (C_{m}⊕ C_{m}) = 2m$ and m is large enough. In particular, it is shown that for each positive integer n there is a positive integer m such that $𝖭₁(C_{mn}⊕ C_{mn}) = 2mn$. Our results partly confirm a conjecture given by W. Narkiewicz thirty years ago, and improve the known results substantially.
5
Content available remote

A quantitative aspect of non-unique factorizations: the Narkiewicz constants III

51%
Gao W., Peng J., Zhong Q.
Acta Arithmetica
|
2013
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tom 158
|
nr 3
271-285
EN
Let K be an algebraic number field with non-trivial class group G and $𝓞_K$ be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let $F_k(x)$ denote the number of non-zero principal ideals $a𝓞_K$ with norm bounded by x such that a has at most k distinct factorizations into irreducible elements. It is well known that $F_k(x)$ behaves for x → ∞ asymptotically like $x(log x)^{1-1/|G|} (log log x)^{𝖭_k (G)}$. We prove, among other results, that $𝖭₁(C_{n₁} ⊕ C_{n₂}) = n₁ + n₂$ for all integers n₁,n₂ with 1 < n₁|n₂.
6
Content available remote

On sequences over a finite abelian group with zero-sum subsequences of forbidden lengths

51%
Gao W., Li Y., Zhao P., Zhuang J.
Colloquium Mathematicum
|
2016
|
tom 144
|
nr 1
31-44
EN
Let G be an additive finite abelian group. For every positive integer ℓ, let $disc_{ℓ}(G)$ be the smallest positive integer t such that each sequence S over G of length |S| ≥ t has a nonempty zero-sum subsequence of length not equal to ℓ. In this paper, we determine $disc_{ℓ}(G)$ for certain finite groups, including cyclic groups, the groups $G = C₂ ⊕ C_{2m}$ and elementary abelian 2-groups. Following Girard, we define disc(G) as the smallest positive integer t such that every sequence S over G with |S| ≥ t has nonempty zero-sum subsequences of distinct lengths. We shall prove that $disc(G) = max{disc_{ℓ}(G) | ℓ ≥ 1}$ and determine disc(G) for finite abelian p-groups G, where p ≥ r(G) and r(G) is the rank of G.
7
Content available remote

On additive bases II

45%
Gao W., Han D., Qian G., Qu Y., Zhang H.
Acta Arithmetica
|
2015
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tom 168
|
nr 3
247-267
EN
Let G be an additive finite abelian group, and let S be a sequence over G. We say that S is regular if for every proper subgroup H ⊆ G, S contains at most |H|-1 terms from H. Let 𝖼₀(G) be the smallest integer t such that every regular sequence S over G of length |S| ≥ t forms an additive basis of G, i.e., every element of G can be expressed as the sum over a nonempty subsequence of S. The constant 𝖼₀(G) has been determined previously only for the elementary abelian groups. In this paper, we determine 𝖼₀(G) for some groups including the cyclic groups, the groups of even order, the groups of rank at least five, and all the p-groups except $G=C_p ⊕ C_{p^n}$ with n≥ 2.
8
Content available remote

On the index of sequences over cyclic groups

45%
Gao W., Li Y., Peng J., Plyley C., Wang G.
Acta Arithmetica
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2011
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tom 148
|
nr 2
119-134
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