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A note on minimal zero-sum sequences over ℤ

Papa A. Sissokho (2014)

Acta Arithmetica

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 , . . . , a h and negative terms b , . . . , b k . We prove that h ≤ ⌊σ⁺/k⌋ and k ≤ ⌊σ⁺/h⌋, where σ = i = 1 h a i = - 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.

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

Weidong Gao, Yuanlin Li, Jiangtao Peng (2011)

Colloquium Mathematicae

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 ( x ) 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 ( x ) behaves, for x → ∞, asymptotically like x ( l o g x ) 1 / | G | - 1 ( l o g l o g x ) k ( G ) . In this article, it is proved that for every prime p, ( C p C p ) = 2 p , and it is also proved that ( C m p C m p ) = 2 m p if ( C m C m ) = 2 m and m is large enough. In particular, it is shown that for...

A structure theorem for sets of small popular doubling

Przemysław Mazur (2015)

Acta Arithmetica

We prove that every set A ⊂ ℤ satisfying x m i n ( 1 A * 1 A ( x ) , t ) ( 2 + δ ) t | A | 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 ( | ( A + A ) | k ) = Θ ( 2 - k / 2 ) .

Inverse zero-sum problems in finite Abelian p-groups

Benjamin Girard (2010)

Colloquium Mathematicae

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...

Kneser’s theorem for upper Banach density

Prerna Bihani, Renling Jin (2006)

Journal de Théorie des Nombres de Bordeaux

Suppose A is a set of non-negative integers with upper Banach density α (see definition below) and the upper Banach density of A + A is less than 2 α . 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 2 α g - 1 arithmetic sequences [We call a set of the form a + d an arithmetic sequence of difference d and call a set of the form { a , a + d , a + 2 d , ... , a + k d } an arithmetic progression of difference d . So an arithmetic progression is finite and an arithmetic sequence...

Large sets with small doubling modulo p are well covered by an arithmetic progression

Oriol Serra, Gilles Zémor (2009)

Annales de l’institut Fourier

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

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