Zero - Sum Problems and Snevily ’ S Conjecture

This is a survey of recent advances on zero-sum problems and Snevily’s conjecture concerning finite abelian groups. In particular, we will introduce Reiher’s recent solution to the Kemnitz conjecture and our simplification. 1. On Zero-sum Problems The theory of zero-sums began with the following celebrated theorem. The Erdős-Ginzburg-Ziv Theorem [Bull. Research Council. Israel, 1961]. For any c1, · · · , c2n−1 ∈ Z, there is an I ⊆ [1, 2n−1] = {1, · · · , 2n− 1} with |I| = n such that ∑ i∈I ci ≡ 0 (mod n). In other words, given 2n−1 (not necessarily distinct) elements of Zn = Z/nZ, we can select n of them with the sum vanishing. The EGZ theorem can be deduced from the well-known ChevalleyWarning theorem or the Cauchy-Davenport theorem. For a finite abelian group G (written additively), the Davenport constant D(G) is defined as the smallest positive integer l such that any 1