Two generalized constants related to zero-sum problems for two special sets.
A characterization of sequences with the minimum number of k-sums modulo k
Let G be an additive abelian group of order k, and S be a sequence over G of length k+r, where 1 ≤ r ≤ k-1. We call the sum of k terms of S a k-sum. We show that if 0 is not a k-sum, then the number of k-sums is at least r+2 except for S containing only two distinct elements, in which case the number of k-sums equals r+1. This result improves the Bollobás-Leader theorem, which states that there are at least r+1 k-sums if 0 is not a k-sum.
Subsequence sums of zero-sum free sequences over finite abelian groups
Let G be a finite abelian group of rank r and let X be a zero-sum free sequence over G whose support supp(X) generates G. In 2009, Pixton proved that for r ≤ 3. We show that this result also holds for abelian groups G of rank 4 if the smallest prime p dividing |G| satisfies p ≥ 13.
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