Name
Abstract | ||
|---|---|---|
Let G≅Z/m1Z×…×Z/mrZ be a finite abelian group with 1<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|. We also generalize Olson's result for |G|-term subsums to an analogous one for n-term subsums when n≥exp(G), with the bound likewise improved for several special classes of groups. |
Year | DOI | Venue |
|---|---|---|
2018 | 10.1016/j.jcta.2018.06.003 | Journal of Combinatorial Theory, Series A |
Keywords | Field | DocType |
Zero-sum,Sumset,Subsequence sum,Subsum,Partition theorem,Kemperman structure theorem,n-fold sumset,Iterated sumset,Olson,Complete sequence | Structured program theorem,Abelian group,Combinatorics,Longest increasing subsequence,Algebra,Generalization,Longest alternating subsequence,Subsequence,Iterated function,Mathematics | Journal |
Volume | ISSN | Citations |
160 | 0097-3165 | 0 |
PageRank | References | Authors |
0.34 | 2 | 1 |
Name | Order | Citations | PageRank |
|---|---|---|---|
| David J. Grynkiewicz | 1 | 42 | 10.33 |