ON TINY ZERO-SUM SEQUENCES OVER FINITE ABELIAN GROUPS

Let G be an additive finite abelian group. Let S = g(1) .... g(l) be a sequence over G, and k(S) = ord(g(1))(-1)+ ... + ord(g(l))(-1) be its cross number. Let t(G) (resp. eta(G)) be the smallest integer t such that every sequence of t elements, repetition allowed, from G has a non-empty zero-sum subsequence T with k(T) <= 1 (resp. vertical bar T vertical bar <= exp(G)). It is easy to see that t(G) >= eta (G). It is known that t(G) = eta(G) = vertical bar G vertical bar when G is cyclic, and for any integer r >= 3 there are infinitely many groups G of rank r such that t(G) > eta (G). Girard (2012) conjectured that t(G) = eta(G) for all finite abelian groups of rank 2. This conjecture has been verified only for the groups G similar or equal to C-p alpha circle plus C-p alpha, G similar or equal to C-2 circle plus C-2p and G similar or equal to C-3 circle plus C-3p with p >= 5, where p is a prime. We confirm this conjecture for more groups, including the groups G similar or equal to C-n circle plus C-n with the smallest prime divisor of n not less than the number of distinct prime divisors of n.