Sumsets, Zero-Sums and Extremal Combinatorics

Citation

Grynkiewicz, David Joseph (2006) Sumsets, Zero-Sums and Extremal Combinatorics. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/M0QN-0V14. https://resolver.caltech.edu/CaltechETD:etd-08102005-190712

Abstract

This thesis develops and applies a method of tackling zero-sum additive questions, especially those related to the Erdos-Ginzburg-Ziv Theorem (EGZ), through the use of partitioning sequences into sets, i.e., set partitions. Much of the research can alternatively be found in the literature spread across nine separate articles, but is here collected into one cohesive work augmented by additional exposition. Highlights include a new combinatorial proof of Kneser's Theorem (not currently located elsewhere); a proof of Caro's conjectured weighted Erdos-Ginzburg-Ziv Theorem; a partition analog of the Cauchy-Davenport Theorem that encompasses several results of Mann, Olson, Bollobas and Leader, and Hamidoune; a refinement of EGZ showing that an essentially dichromatic sequence of 2m-1 terms from an abelian group of order m contains a mostly monochromatic m-term zero-sum subsequence; an interpretation of Kemperman's Structure Theorem (KST) for critical pairs (i.e., those finite subsets A and B of an abelian group with |A+B|<|A|+|B|) through quasi-periodic decompositions, which establishes certain canonical aspects of KST and facilitates its use in practice; a draining theorem for set partitions showing that a set partition of large cardinality sumset can have several elements removed from its terms and still have the sumset remain of large cardinality; a proof of a subsequence sum conjecture of Hamidoune; the determination of the g(m,k) function introduced by Bialostocki and Lotspeich (defined as the least n so that a sequence of terms from Z/mZ of length n with at least k distinct terms must contain an m-term zero-sum subsequence) for m large with respect to k; the determination of g(m,5) for all m, including the details to the abbreviated proof found in the literature; various zero-sum results concerning modifications to the nondecreasing diameter problem of Bialostocki, Erdos, and Lefmann; an extension of EGZ to a class of hypergraphs; and a lower bound on the number of zero-sum m-term subsequences in a sequence of n terms from an abelian group of order m that establishes Bialostocki's conjectured value for small n<(19/3)m .

Thesis Files