This short survey, based on the content of a talk with the same title, summarizes some classical and recent results on the set of differences of an abelian group. We put a certain emphasize on ongoing joint work of A. Plagne and the author. We also briefly review the relevance of this notion in Non-unique Factorization Theory, in particular towards the subject mentioned in the title.
@article{ACIRM_2010__2_2_103_0, author = {Wolfgang A. Schmid}, title = {Towards a more precise understanding of sets of lengths}, journal = {Actes des rencontres du CIRM}, pages = {103--105}, publisher = {CIRM}, volume = {2}, number = {2}, year = {2010}, doi = {10.5802/acirm.43}, zbl = {06938591}, language = {en}, url = {https://acirm.centre-mersenne.org/articles/10.5802/acirm.43/} }
TY - JOUR AU - Wolfgang A. Schmid TI - Towards a more precise understanding of sets of lengths JO - Actes des rencontres du CIRM PY - 2010 SP - 103 EP - 105 VL - 2 IS - 2 PB - CIRM UR - https://acirm.centre-mersenne.org/articles/10.5802/acirm.43/ DO - 10.5802/acirm.43 LA - en ID - ACIRM_2010__2_2_103_0 ER -
Wolfgang A. Schmid. Towards a more precise understanding of sets of lengths. Actes des rencontres du CIRM, Volume 2 (2010) no. 2, pp. 103-105. doi : 10.5802/acirm.43. https://acirm.centre-mersenne.org/articles/10.5802/acirm.43/
[1] L. Carlitz. A characterization of algebraic number fields with class number two. Proc. Amer. Math. Soc., 11:391–392, 1960. | DOI | MR | Zbl
[2] S. Chang, S. T. Chapman, and W. W. Smith. On minimum delta set values in block monoids over cyclic groups. Ramanujan J., 14(1):155–171, 2007. | DOI | MR | Zbl
[3] S. T. Chapman and W. W. Smith. Factorization in Dedekind domains with finite class group. Israel J. Math., 71(1):65–95, 1990. | DOI | MR | Zbl
[4] S. T. Chapman, W. A. Schmid, and W. W. Smith. On minimum distances in Krull monoids with infinite class group. Bull. Lond. Math. Soc., 40(4):613–618, 2008. | DOI | MR | Zbl
[5] G. Freiman and A. Geroldinger. An addition theorem and its arithmetical application. J. Number Theory, 85(1):59–73, 2000. | DOI | MR | Zbl
[6] W. Gao and A. Geroldinger. Systems of sets of lengths. II. Abh. Math. Sem. Univ. Hamburg, 70:31–49, 2000. | DOI | MR | Zbl
[7] A. Geroldinger. Über nicht-eindeutige Zerlegungen in irreduzible Elemente. Math. Z., 197(4):505–529, 1988. | DOI | Zbl
[8] A. Geroldinger and F. Halter-Koch. Non-unique factorizations. Algebraic, Combinatorial and Analytic Theory. Chapman & Hall/CRC, 2006. | DOI | Zbl
[9] A. Geroldinger and Y. ould Hamidoune. Zero-sumfree sequences in cyclic groups and some arithmetical application. J. Théor. Nombres Bordeaux, 14(1):221–239, 2002. | DOI | MR | Zbl
[10] A. Plagne and W. A. Schmid. On congruence half-factorial Dedekind domains with cyclic class group. Manuscript in progress. | DOI | MR | Zbl
[11] W. A. Schmid. Arithmetical characterization of class groups of the form via the system of sets of lengths. Abh. Math. Sem. Hamburg, 79:25–35, 2009. | DOI | MR
Cited by Sources: