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Systems of sets of lengths. II. (English) Zbl 1036.11054

Let \(G\) be a finite additive abelian group of exponent \(n\). For a subset \(G_0 \subset G\), let \(\mathcal B (G_0)\) be the set of all (finite) sequences in \(G_0\) which sum to zero. For \(B \in \mathcal B (G_0)\), let \(L(B)\) be the set of all \(r \in \mathbb N\) such that \(B = B_1 \cdot\ldots\cdot B_r\) with irreducible \(B_i \in \mathcal B(G_0)\). For a finite set \(L = \{a_1, \ldots, a_k\} \subset \mathbb Z\) with \(a_1 < a_2 < \ldots < a_k\), set \(\Delta (L) = \{a_i - a_{i-1} \mid 2 \leq i \leq k\}\), and in particular \(\Delta (L) = \emptyset\) if \(k = 1\). Let \(\Delta (G_0)\) be the union of all sets \(\Delta (L(B))\), where \(B \in \mathcal B (G_0)\), and \[ \Delta^* (G) = \{ \min \Delta(G_0) \mid G_0 \subset G, \;\Delta (G_0) \neq \emptyset \} \,. \] \(\Delta^* (G)\) is a key invariant in the study of non-unique factorizations in Krull monoids with finite class group (and thus in rings of integers in finite algebraic number fields); see the survey articles in [D. D. Anderson (ed.), Factorization in integral domains, Lecture Notes in Pure and Applied Mathematics 189, Marcel Dekker, New York (1997; Zbl 0865.00039)].
The authors prove the following results: 1. If \(| G| \leq n^2\) or \(G\) is a \(p\)-group with “small” rank, then \(\max \Delta^* (G) = n-2\); 2. If \(G\) is cyclic, then \(\max (\Delta^* (G) \setminus \{n-2\}) < \frac{n+1}{2}\) (the smaller bound \(\lfloor \frac{n}{2} \rfloor -1\) was obtained recently by A. Geroldinger and Y. O. Hamidoune [J. Théor. Nombres Bordx. 14, 221–239 (2002; Zbl 1018.11011)]); 3. If \(G\) is a \(p\)-group of “large” rank \(r\), then \(\Delta^*(G) = \{1,2,\ldots, r-1\}\).
The proofs use subtle information concerning the infrastructure of zero-sum sequences and relations with various other invariants studied in additive group theory.

MSC:

11R27 Units and factorization
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
20D60 Arithmetic and combinatorial problems involving abstract finite groups
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