The Independence Number of a Subset of an Abelian Group

We call a subset A of the (additive) abelian group G t-independent if for all non-negative integers h and k with h + k ≤ t, the sum of h (not necessarily distinct) elements of A does not equal the sum of k (not necessarily distinct) elements of A unless h = k and the two sums contain the same terms in some order. A weakly t-independent set satisfies this property for sums of distinct terms. We give some exact values and asymptotic bounds for the size of a largest t-independent set and weakly t-independent set in abelian groups, particularly in the cyclic group Zn.