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On the index of length four minimal zero-sum sequences

100%

Shen C., Xia L.-m., Li Y.

Colloquium Mathematicum

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2014

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tom 135

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nr 2

201-209

EN

Let G be a finite cyclic group. Every sequence S over G can be written in the form $S = (n₁g)·...·(n_{l}g)$ where g ∈ G and $n₁,..., n_{l}i ∈ [1,ord(g)]$, and the index ind(S) is defined to be the minimum of $(n₁+ ⋯ +n_{l})/ord(g)$ over all possible g ∈ G such that ⟨g⟩ = G. A conjecture says that every minimal zero-sum sequence of length 4 over a finite cyclic group G with gcd(|G|,6) = 1 has index 1. This conjecture was confirmed recently for the case when |G| is a product of at most two prime powers. However, the general case is still open. In this paper, we make some progress towards solving the general case. We show that if G = ⟨g⟩ is a finite cyclic group of order |G| = n such that gcd(n,6) = 1 and S = (x₁g)·(x₂g)·(x₃g)·(x₄g) is a minimal zero-sum sequence over G such that x₁,...,x₄ ∈ [1,n-1] with gcd(n,x₁,x₂,x₃,x₄) = 1, and $gcd(n,x_{i}) > 1$ for some i ∈ [1,4], then ind(S) = 1. By using a new method, we give a much shorter proof to the index conjecture for the case when |G| is a product of two prime powers.

2

On a conjecture of Lemke and Kleitman

100%

Zeng X., Li Y., Yuan P.

Acta Arithmetica

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2015

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tom 168

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nr 3

289-299

EN

Let G be a finite cyclic group of order n ≥ 2. Every sequence S over G can be written in the form $S = (n_1 g)· ... · (n_lg)$ where g ∈ G and $n_1, ..., n_l ∈ [1,ord(g)]$, and the index ind(S) of S is defined as the minimum of $(n_1 + ⋯ + n_l )/ord(g)$ over all g ∈ G with ord(g) = n. In this paper it is shown that any sequence S over G of length |S| ≥ n ≥ 5, 2 ∤ n, having an element with multiplicity at least n/3 has a subsequence T with ind(T) = 1. On the other hand, if n,d ≥ 2 are positive integers with d|n and $n > d²(d³-d²+d+1), we provide an example of a sequence S of length |S| ≥ n having an element with multiplicity l = n/d - d(d-1) - 1 such that S has no subsequence T with ind(T) = 1, giving a general counterexample to a conjecture of Lemke and Kleitman.

3

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

100%

Gao W., Li Y., Peng J.

Colloquium Mathematicum

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2011

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tom 124

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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.

4

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

100%

Gao W., Li Y., Zhao P., Zhuang J.

Colloquium Mathematicum

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2016

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tom 144

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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.

5

On the index of sequences over cyclic groups

88%

Gao W., Li Y., Peng J., Plyley C., Wang G.

Acta Arithmetica

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2011

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tom 148

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nr 2

119-134

Ograniczanie wyników

3 Colloquium Mathematicum

2 Acta Arithmetica

5 Li Y.

3 Gao W.

2 Peng J.

1 Plyley C.

1 Shen C.

1 Wang G.

1 Xia L.-m.

1 Yuan P.

1 Zeng X.

1 Zhao P.

1 Zhuang J.

1 2016

1 2015

1 2014

2 2011

On the index of length four minimal zero-sum sequences

On a conjecture of Lemke and Kleitman

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

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

On the index of sequences over cyclic groups