×

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] N. Alon andM. Dubiner, Zero-sum sets of prescribed size.Combinatorics, Paul Erdös is Eighty, Vol. 1. J. Bolyai Math. Soc, (1993), 33–50. · Zbl 0823.11006
[2] D. D. Anderson,Factorization in integral domains. Marcel Dekker, 1997. · Zbl 0865.00039
[3] J. D. Bovey, P. Erdös, andI. Niven, Conditions for zero sum modulon. Canad. Math. Bull 18 (1975), 27–29. · Zbl 0314.10040 · doi:10.4153/CMB-1975-004-4
[4] L. Carlitz, A characterization of algebraic number fields with class number two.Proc. AMS 11 (1960), 391–392. · Zbl 0202.33101
[5] S. Chapman andA. Geroldinger, Krull domains and monoids, their sets of lengths and associated combinatorial problems.Factorization in integral domains. Lecture Notes in Pure Appl. Math. 189. Marcel Dekker (1997), 73–112. · Zbl 0897.13001
[6] P. Erdös andH. Heilbronn, On the addition of residue classes modulop. Acta Arith. 9 (1964), 149–159.
[7] A. Geroldinger, On non-unique factorizations into irreducible elements II.Colloquia Mathematica Societatis Janos Bolyai 51. North Holland (1987), 723–757.
[8] –, Über nicht-eindeutige Zerlegungen in irreduzible Elemente.Math. Z. 197 (1988), 505–529. · Zbl 0618.12002 · doi:10.1007/BF01159809
[9] –, Systeme von Längenmengen.Abh. Math. Sem. Univ. Hamburg 60 (1990), 115–130. · Zbl 0721.11042 · doi:10.1007/BF02941052
[10] –, The cross number of finite abelian groups.J. Number Theory 48 (1994), 219–223. · Zbl 0814.20033 · doi:10.1006/jnth.1994.1063
[11] –, A structure theorem for sets of lengths.Colloq. Math. 78 (1998), 225–259. · Zbl 0926.11082
[12] W. Gao andA. Geroldinger, Half-factorial domains and half-factorial subsets in abelian groups.Houston J. Math. 24 (1998), 593–611. · Zbl 0994.20046
[13] –, On long minimal zero sequences in finite abelian groups.Periodica Math. Hungarica 38 (1999), 179–211. · Zbl 0980.11014 · doi:10.1023/A:1004854508165
[14] –, On the structure of zerofree sequences.Combinatorica 18 (1998), 519–527. · Zbl 0968.11016 · doi:10.1007/s004930050037
[15] A. Geroldinger andR. Schneider, The cross number of finite abelian groups II.European J. Combinatorics 15 (1994), 399–405. · Zbl 0833.20061 · doi:10.1006/eujc.1994.1043
[16] –, The cross number of finite abelian groups III.Discrete Math. 150 (1996), 123–130. · Zbl 0848.20048 · doi:10.1016/0012-365X(95)00181-U
[17] W. Gao andY.X. Yang, Note on a combinatorial constant.J. Math. Res. and Expo. 17 (1997), 139–140. · Zbl 0895.20045
[18] H. Harborth, Ein Extremalproblem für Gitterpunkte.J. Reine Angew. Math. 262 (1973), 356–360. · Zbl 0268.05008 · doi:10.1515/crll.1973.262-263.356
[19] F. Halter-Koch, Über Längen nicht-eindeutiger Faktorisierungen und Systeme linearer diopantischer Ungleichungen.Abh. Math. Sem. Univ. Hamburg 63 (1993), 265–276. · Zbl 0797.20055 · doi:10.1007/BF02941346
[20] Y.O. Hamidoune andG. Zémor, On zero-free subset sums.Acta Arith. 78 (1996), 143–152.
[21] F. Kainrath, Factorization in Krull monoids with infinite class group.Colloq. Math. 80 (1999), 23–30. · Zbl 0936.20050
[22] A. Kemnitz, On a lattice point problem.Ars Combinatoria 16 (1983), 151–160. · Zbl 0539.05008
[23] W. Narkiewicz,Elementary and Analytic theory of algebraic numbers. Springer, 1990. · Zbl 0717.11045
[24] J.E. Olson, Sums of sets of group elements.Acta Arith. 28 (1975), 147–156. · Zbl 0318.10035
[25] J. Sliwa, Remarks on factorizations in algebraic number fields.Coll. Math. 46 (1982), 123–130. · Zbl 0514.12005
[26] P. van Emde Boas, A combinatorial problem on finite abelian groups II.Reports ZW-1969-007. Math. Centre, Amsterdam, 1969. · Zbl 0203.32703
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.