Let n be an integer, and consider finite sequences of elements of the group
Z/nZ x Z/nZ. Such a sequence is called zero-sum free, if no subsequence has sum
zero. It is known that the maximal length of such a zero-sum free sequence is
2n-2, and Gao and Geroldinger conjectured that every zero-sum free sequence of
this length contains an element with multiplicity at least n-2. By recent
results of Gao, Geroldinger and Grynkiewicz, it essentially suffices to verify
the conjecture for n prime. Now fix a sequence (a_i) of length 2n-2 with
maximal multiplicity of elements at most n-3. There are different approeaches
to show that (a_i) contains a zero-sum; some work well when (a_i) does contain
elements with high multiplicity, others work well when all multiplicities are
small. The aim of this article is to initiate a systematic approach to property
B via the highest occurring multiplicities. Our main results are the following:
denote by m_1 >= m_2 the two maximal multiplicities of (a_i), and suppose that
n is sufficiently big and prime. Then (a_i) contains a zero-sum in any of the
following cases: when m_2 >= 2/3n, when m_1 > (1-c)n, and when m_2 < cn, for
some constant c > 0 not depending on anything.Comment: 27 pages, 3 figure