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

Q2 Mathematics
David J. Grynkiewicz
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引用次数: 0

Abstract

Let GZ/m1Z××Z/mrZ be a finite abelian group with m1||mr=exp(G). The Kemperman Structure Theorem characterizes all subsets A,BG 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 AG with |nA|(|A|+1)n3 when n3 (also when G is infinite), in which case many of the pathological possibilities from the case n=2 vanish, particularly for large nexp(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 HG. 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|.

迭代的集合和Olson对Erdős-Ginzburg-Ziv定理的推广
设G = Z/ m1zx…×Z/mrZ是一个有限阿贝尔群,其中m1|…|mr=exp (G)。kempman结构定理描述了所有子集A、B的≥≥A+B; <|A|+|B|,并将其扩展到≤|A|+|B|的情况。利用这些结果,我们提供了当n≥3(也当G为无限大)时,所有具有|nA|≤(| a |+1)n−3的有限子集a的精确结构描述,在这种情况下,当n=2时的许多病态可能性消失,特别是当n≥exp (G)−1时。该结构描述与其他论证相结合,推广了Olson的子序列和结果,即由长度为|S|≥2|G|−1的G的项组成的序列S,要么G的每一个元素都可以表示为来自S的|G|项的和,要么除了|G/H|−2项以外的所有项都在H-余集内,对于某些H≤G。我们证明了弱得多的假设|S|≥|G|+exp (G)足以得到一个几乎相同的结论,其中对于H是平凡的情况,我们必须允许S的除|G/H|−1项以外的所有项都来自同一个H集。改进了若干类群G上的界,得到了最优下界。
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来源期刊
Electronic Notes in Discrete Mathematics
Electronic Notes in Discrete Mathematics Mathematics-Discrete Mathematics and Combinatorics
CiteScore
1.30
自引率
0.00%
发文量
0
期刊介绍: Electronic Notes in Discrete Mathematics is a venue for the rapid electronic publication of the proceedings of conferences, of lecture notes, monographs and other similar material for which quick publication is appropriate. Organizers of conferences whose proceedings appear in Electronic Notes in Discrete Mathematics, and authors of other material appearing as a volume in the series are allowed to make hard copies of the relevant volume for limited distribution. For example, conference proceedings may be distributed to participants at the meeting, and lecture notes can be distributed to those taking a course based on the material in the volume.
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