The purpose of this series of lectures is to give an overview of zero-sum problems starting from classical results to questions of current research. A basic problem in zero-sum theory is the following problem: given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a non-empty subsequence whose terms sum to 0. This l is called the Davenport constant of G, denoted D(G). A classical variant of this problem is obtained by imposing a condition on the length of the subsequences: given a finite abelian group (G, +, 0) what is the smallest l such that each sequence of l elements over G has a subsequence of length exp(G) whose terms sum to 0. This l is called the Erd ̋os–Ginzburg–Ziv constant of G, denoted s(G).

The value of both constants is known for groups of rank at most two, yet not for general groups of rank three or higher.

A first goal is to present the proofs for groups of rank at most two. There are two main parts. First, arguments based on counting constellations of zero-sum subsequences, which typically are obtained via group-ring or polynomial methods. These yield result for p-groups. Second, inductive arguments, which allow to obtain insights on zero-sum constants for a group G from knowledge of zero-sum constants for a subgroup H and the quotient group G/H. A second goal is to review further results to obtain a good overview of what

is currently known on these constants.

Additional subjects to be discussed are the analogous problems for sets in-stead of sequences, the analogous problems for weighted sequences, and (time permitting) some applications.

Tentative schedule and selected references:

1. Introduction and overview

2. Davenport constant for p-groups, related results (towards a proof of Kemnitz conjecture)

3. Proof of Kemnitz conjecture and inductive arguments

4. Some recent advances on zero-sum constants

5. Problems with weights

6. Set-based problems (Olson constant, Harborth constant)

7. Applications, summary, perspectives.

Selected references:

1. S. D. Adhikari, Plus-minus weighted zero-sum constans: a survey. In: Analytic Number Theory, Modular Forms and q-Hypergeometric Series (G. E. Andrews and F. Garvan, eds), Springer 2018.

2. W. D. Gao and A. Geroldinger, Zero-sum problems in finite abelian groups: a survey. Expo. Math. 24 (2006), 337–369.

3. A. Geroldinger, Additive group theory and non-unique factorizations. In: Combinatorial Number Theory and Additive Group Theory (A. Geroldinger and I. Z. Ruzsa, eds), Birkh ̈auser 2009.