2017/09/27 by Grynkiewicz, David J.
#11B75 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1709.09288
Let G≅ \mathbb Z/m1\mathbb Z×…× \mathbb Z/mr\mathbb Z be a finite abelian group with m1|…| mr=exp(G). The n-term subsums version of Kneser's Theorem, obtained either via the DeVos-Goddyn-Mohar Theorem or the Partition Theorem, has become a powerful tool used to prove numerous zero-sum and subsequence sum questions. It provides a structural description of sequences having a small number of n-term subsequence sums, ensuring this is only possible if most terms of the sequence are contained in a small number of H-cosets. For large n≥ \frac1p|G|-1 or n≥ \frac1p|G|+p-3, where p is the smallest prime divisor of |G|, the structural description is particularly strong. In particular, most terms of the sequence become contained in a single H-coset, with additional properties holding regarding the representation of elements of G as subsequence sums. This strengthened form of the subsums version of Kneser's Theorem was later to shown to hold under the weaker hypothesis n≥ \mathsf d^*(G), where \mathsf d^*(G)=∑i=1r(mi-1). In this paper, we reduce the restriction on n even further to an optimal, best-possible value, showing we need only assume n≥ exp(G)+1 to obtain the same conclusions, with the bound further improved for several classes of near-cyclic groups.