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The cardinality of sumsets: different summands

Brendan Murphy, Eyvindur Ari Palsson, Giorgis Petridis (2015)

Acta Arithmetica

We offer a complete answer to the following question on the growth of sumsets in commutative groups. Let h be a positive integer and A , B , . . . , B h be finite sets in a commutative group. We bound | A + B + . . . + B h | from above in terms of |A|, |A + B₁|, ..., | A + B h | and h. Extremal examples, which demonstrate that the bound is asymptotically sharp in all parameters, are furthermore provided.

The Davenport constant of a box

Alain Plagne (2015)

Acta Arithmetica

Given an additively written abelian group G and a set X ⊆ G, we let (X) denote the monoid of zero-sum sequences over X and (X) the Davenport constant of (X), namely the supremum of the positive integers n for which there exists a sequence x₁⋯xₙ in (X) such that i I x i 0 for each non-empty proper subset I of 1,...,n. In this paper, we mainly investigate the case when G is a power of ℤ and X is a box (i.e., a product of intervals of G). Some mixed sets (e.g., the product of a group by a box) are studied...

The set of minimal distances in Krull monoids

Alfred Geroldinger, Qinghai Zhong (2016)

Acta Arithmetica

Let H be a Krull monoid with class group G. Then every nonunit a ∈ H can be written as a finite product of atoms, say a = u 1 · . . . · u k . The set (a) of all possible factorization lengths k is called the set of lengths of a. If G is finite, then there is a constant M ∈ ℕ such that all sets of lengths are almost arithmetical multiprogressions with bound M and with difference d ∈ Δ*(H), where Δ*(H) denotes the set of minimal distances of H. We show that max Δ*(H) ≤ maxexp(G)-2,(G)-1 and that equality holds if every...

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