# Inverse zero-sum problems

@article{Gao2007InverseZP, title={Inverse zero-sum problems}, author={Weidong Gao and Alfred Geroldinger and Wolfgang A. Schmid}, journal={Acta Arithmetica}, year={2007}, volume={128}, pages={245-279} }

Let G be an additive finite abelian group with exponent exp(G) = n. We define some central invariants in zero-sum theory: Let • D(G) denote the smallest integer l ∈ N such that every sequence S over G of length |S| ≥ l has a zero-sum subsequence. • η(G) denote the smallest integer l ∈ N such that every sequence S over G of length |S| ≥ l has a zero-sum subsequence T of length |T | ∈ [1, n]. • s(G) denote the smallest integer l ∈ N such that every sequence S over G of length |S| ≥ l has a zero…

## 52 Citations

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Let K be an algebraic number field with non-trivial class group G and let be its ring of integers. For k ∈ ℕ and some real x ≥ 1, let Fk (x) denote the number of non-zero principal ideals with norm…

## References

SHOWING 1-10 OF 35 REFERENCES

### Zero-sum problems in finite abelian groups and affine caps

- Mathematics
- 2006

For a finite abelian group G, let (G) denote the smallest integer l such that every sequence S over G of length | S| l has a zero-sum subsequence of length exp (G). We derive new upper and lower…

### Sequences in abelian groups G of odd order without zero-sum subsequences of length exp(G)

- MathematicsDes. Codes Cryptogr.
- 2008

A new construction for sequences in the finite abelian group C_{n}^r without zero-sum subsequences of length n, for odd n is presented, which improves the maximal known cardinality of such sequences for r > 4 and leads to simpler examples for r < 2.

### Two Zero-Sum Problems and Multiple Properties☆

- Mathematics
- 2000

Abstract In this paper we consider the following open problems: Conjecture 0.1. Let S be a sequence of 3 n −3 elements in C n ⊕ C n . If S contains no nonempty zero-sum subsequence of length not…

### On some developments of the Erdős–Ginzburg–Ziv Theorem II

- Mathematics
- 2003

Let S be a sequence of elements from the cyclic group Zm. We say S is zsf (zero-sum free) if there does not exist an m-term subsequence of S whose sum is zero. Denote by g(m, k) the least integer…

### A combinatorial problem in Abelian groups

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 1963

Let α be a prime element of the ring of integers of an algebraic number field, R. Mr C. Sudler verbally raised the question as to how many prime ideal factors α can have. This is equivalent to a…