Iterated Sumsets and Subsequence Sums
David J. Grynkiewicz

TL;DR
This paper characterizes the structure of subsets and sequences in finite abelian groups with specific sumset and subsequence sum properties, extending classical theorems and improving bounds for various group classes.
Contribution
It provides a precise structural description of subsets with small multiple sumsets and generalizes Olson's subsequence sum results with optimal bounds.
Findings
Structural description of subsets with small n-fold sumsets
Generalization of Olson's subsequence sum theorem with weaker hypotheses
Improved bounds for specific classes of groups
Abstract
Let be a finite abelian group with . The Kemperman Structure Theorem characterizes all subsets satisfying and has been extended to cover the case when . Utilizing these results, we provide a precise structural description of all finite subsets with when (also when is infinite), in which case many of the pathological possibilities from the case vanish, particularly for large . The structural description is combined with other arguments to generalize a subsequence sum result of Olson asserting that a sequence of terms from having length must either have every element of representable as a sum of -terms from or else have all…
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