Iterated Sumsets and Olson's Generalization of the Erdős-Ginzburg-Ziv Theorem


Let G≅Z/m1Z×…×Z/mrZ be a finite abelian group with m1|…|mr=exp⁡(G). The Kemperman Structure Theorem characterizes all subsets A,B⊆G satisfying |A+B|<|A|+|B| and has been extended to cover the case when |A+B|≤|A|+|B|. Utilizing these results, we provide a precise structural description of all finite subsets A⊆G with |nA|≤(|A|+1)n−3 when n≥3 (also when G is infinite), in which case many of the pathological possibilities from the case n=2 vanish, particularly for large n≥exp⁡(G)−1. The structural description is combined with other arguments to generalize a subsequence sum result of Olson asserting that a sequence S of terms from G having length |S|≥2|G|−1 must either have every element of G representable as a sum of |G|-terms from S or else have all but |G/H|−2 of its terms lying in a common H-coset for some H≤G. We show that the much weaker hypothesis |S|≥|G|+exp⁡(G) suffices to obtain a nearly identical conclusion, where for the case H is trivial we must allow all but |G/H|−1 terms of S to be from the same H-coset. The bound on |S| is improved for several classes of groups G, yielding optimal lower bounds for |S|.

Publication Title

Electronic Notes in Discrete Mathematics