Starting Date: 1 July, 2010

Ending Date: 30 September, 2013

Karl-Franzens-Universität Graz

Institut für Mathematik und Wissenschaftliches Rechnen

Let G be an abelian group, let A and B be finite, nonempty subsets, and let A+B = {a+b | a lies in A and b in B } denote their sumset. Direct addition theorems study the size of the sumset in terms of |A| and |B|. For instance, Kneser’s Addition Theorem (proved in the 1950s) states that |A+B| is at least |A+H| + |B+H| - |H|, where H = {h from G | h+A+B=A+B} denotes the stabilizer of A+B. Inverse addition theorems (such as the Theorems of Vosper, Kemperman, and Freiman) give information on the structure of A and B under an assumption that |A+B| is small.

This field has its origin in Combinatorial
Number Theory. Its objective are (finite) sequences S = g_1 …
g_l over an abelian group G (where in S, the repetition of
elements is allowed and their order is disregarded). We say S
has sum zero if g_1+ …+g_l = 0. A typical direct zero-sum
problem studies conditions which ensure that given sequences
have nontrivial zero-sum subsequences with prescribed
properties. Zero-sum theory is closely connected with various
branches of combinatorics, graph theory and geometry. The Erdös-Ginzburg-Ziv
Theorem (proved in 1961) is considered as one of the
starting points of the area. It states that a sequence S over
a finite cyclic group with length |S| at least 2|G|-1
has a zero-sum subsequence of length |G|. It was only in 2007
that this result found its final generalization
to groups of rank two.

Summary

The present project lies in the intersection of the above mentioned areas. Apart from polynomial methods and group algebras, addition theorems are a central tool in zero-sum theory. If the noetherian domain R considered above is integrally closed, then it is a Krull domain, and arithmetical questions in R can be translated into zero-sum problems over its class group G. This transfer process gives optimal results when the class group is finite and every class contains prime divisors (all these assumptions are satisfied for rings of integers in algebraic number fields). A goal of the present project is to apply recent progress in inverse addition theorems and in inverse zero-sum problems to study which differences are possible for sets of lengths (recall that these sets of lengths are large subsets of (generalized) arithmetical progressions).

Summary

The present project lies in the intersection of the above mentioned areas. Apart from polynomial methods and group algebras, addition theorems are a central tool in zero-sum theory. If the noetherian domain R considered above is integrally closed, then it is a Krull domain, and arithmetical questions in R can be translated into zero-sum problems over its class group G. This transfer process gives optimal results when the class group is finite and every class contains prime divisors (all these assumptions are satisfied for rings of integers in algebraic number fields). A goal of the present project is to apply recent progress in inverse addition theorems and in inverse zero-sum problems to study which differences are possible for sets of lengths (recall that these sets of lengths are large subsets of (generalized) arithmetical progressions).

Zero-Sum Problems with
Congruence Conditions A. Geroldinger, D. J. Grynkiewicz, and W. A. Schmid |
Acta Math. Hungar. 131 (2011), 323-345. pdf |

The Catenary Degree of
Krull Monoids I A. Geroldinger, D. J. Grynkiewicz and W. A. Schmid |
J. Théor. Nombres
Bordeaux 23 (2011), 137-169. pdf |

On the Arithmetic of
Tame Monoids with Applications to Krull Monoids and
Mori Domains A. Geroldinger and F. Kainrath |
J. Pure and Applied
Algebra 214 (2010), 2199-2218. pdf |

A Quantitative Aspect of
Non-Unique Factorizations: The Narkiewicz Constants W. D. Gao, A. Geroldinger, and Q. Wang |
Int. J. Number Theory 7 (2011), 1463-1502. pdf |

On the Davenport
Constant and on the Structure of Extremal Zero-Sum
Free Sequences A. Geroldinger, M. Liebmann, and A. Philipp |
Period. Math. Hungar. 64 (2012). pdf |

Semigroup-Theoretical
Characterizations of Arithmetical Invariants with
Applications to Numerical Monoids and Krull Monoids Victor Blanco, Pedro A. García-Sánchez, and A. Geroldinger |
Illinois J. Math, 55 (2011), 1385 - 1414. pdf |

On Weighted Zero-Sum
Sequences S. Adhikari, D. J. Grynkiewicz, and Z. W. Sun |
Adv. Appl. Math. 48 (2012), 506-527. pdf |

Inverse Additive Problems for
Minkowski Sumsets I G. A. Freiman, D. J. Grynkiewicz, O. Serra and Y. Stanchescu |
Collectanea Mathematica, 63 (2012), 261-286. pdf |

Inverse Additive Problems for
Minkowski Sumsets II G. A. Freiman, D. J. Grynkiewicz, O. Serra and Y. Stanchescu |
J. Convex Analysis 23 (2013), 395-414. pdf |

Structure of general ideal semigroups
of monoids and domains A. Reinhart |
Journal
of Commutative Algebra 4 (2012), 413-444. pdf |

Arithmetic Progression Weighted
Subsequence Sums D. J. Grynkiewicz, A. Philipp, and V. Ponomarenko |
Israel Journal of
Mathematics 193 (2013), 359-398. pdf |

Products of Two Atoms in
Krull Monoids and Arithmetical Characterization of
Class Groups P. Baginski, A. Geroldinger, D. J. Grynkiewicz, and A. Philipp |
Europ.
J. Comb., 34 (2013), 1244-1268. pdf |

On
integral domains that are C-monoids A. Reinhart |
Houston J. Math to appear pdf |

Non-Commutative
Krull Monoids: A Divisor Theoretic Approach and A. Geroldinger |
Osaka J. Math., 50 (2013), 503-539. pdf |

The
Monotone Catenary Degree of Krull Monoids A. Geroldinger and P. Yuan |
Results
in Mathematics, 63 (2013), 999-1031. pdf |

The
Set of Distances in Krull Monoids A. Geroldinger and P. Yuan |
Bull.
Lond. Math. Soc, 44 (2012), 1203 -- 1208, pdf |

Radical
factorial monoids and related conceptsA. Reinhart |
Annales
des sciences mathématiques du Québec, 36 (2012), 193-229. pdf |

The Large Davenport
Constant I: Groups with a Cyclic, Index 2 SubgroupA. Geroldinger and D. J. Grynkiewicz |
J. Pure and Applied
Algebra, 217 (2013), 863 - 885, pdf |

The Large Davenport
Constant II: General Upper Bounds D. J. Grynkiewicz |
J. Pure and Applied
Algebra, 217 (2013), 2221-2246. pdf |

Local and Global
Tamesness in Krull MonoidsW. D. Gao, A. Geroldinger, and W. A. Schmid |
Communications in
Algebra, to appear. pdf |

Monoids of Modules
and Arithmetic of Direct-Sum Decompositions N. R. Baeth and A. Geroldinger |
Manuscript, pdf |

On Monoids and
Domains whose Monadic Submonoids are Krull A. Reinhart |
to appear in Commutative Rings,
Integer-Valued Polynomials and Polynomial Functoins,
Springer, eds. M. Fontana, S. Frisch, and S. Glaz, pdf |

Classification of Monomial p-Central
Spaces in Central Simple Algebras of Degree p A. Chapman, D. J. Grynkiewicz, E. Matrzi, L Rowen, and U. Vishne |
Manuscript |

Structural
Additive Theory D. J. Grynkiewicz |
Developments
in Mathematics 30, Springer (June 2013).Springer Link |

Last
updated 12.11.2013

David J. Grynkiewicz

Priv. Doz. Dr.

Karl-Franzens-Universität Graz

Institut für Mathematik und Wissenschaftliches Rechnen

Heinrichstrasse 36, 8010 Graz, Austria, Room 528

+ 43 316 380 5155 (Office)

+ 43 316 380 9815 (Fax)

diambri@hotmail.com

Priv. Doz. Dr.

Karl-Franzens-Universität Graz

Institut für Mathematik und Wissenschaftliches Rechnen

Heinrichstrasse 36, 8010 Graz, Austria, Room 528

+ 43 316 380 5155 (Office)

+ 43 316 380 9815 (Fax)

diambri@hotmail.com

Andreas Reinhart

Dr. Mag.

Karl-Franzens-Universität Graz

Institut für Mathematik und Wissenschaftliches Rechnen

Heinrichstrasse 36, 8010 Graz, Austria, Room 524

+ 43 316 380 5150 (Office)

+ 43 316 380 9815 (Fax)

andreas.reinhart@uni-graz.at

Dr. Mag.

Karl-Franzens-Universität Graz

Institut für Mathematik und Wissenschaftliches Rechnen

Heinrichstrasse 36, 8010 Graz, Austria, Room 524

+ 43 316 380 5150 (Office)

+ 43 316 380 9815 (Fax)

andreas.reinhart@uni-graz.at

Alfred Geroldinger

Ao. Univ. Prof. Dr. Dipl.-Ing Mag.

Karl-Franzens-Universität Graz

Institut für Mathematik und Wissenschaftliches Rechnen

Heinrichstrasse 36, 8010 Graz, Austria, Room 526

+ 43 316 380 5154 (Office)

+ 43 316 380 9815 (Fax)

alfred.geroldinger@uni-graz.at

Ao. Univ. Prof. Dr. Dipl.-Ing Mag.

Karl-Franzens-Universität Graz

Institut für Mathematik und Wissenschaftliches Rechnen

Heinrichstrasse 36, 8010 Graz, Austria, Room 526

+ 43 316 380 5154 (Office)

+ 43 316 380 9815 (Fax)

alfred.geroldinger@uni-graz.at