Iterated sumsets and setpartitions

Let G≅ Z/ m1Z× ⋯ × Z/ mrZ 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≥1p|G|-1 or n≥1p|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 subsum version of Kneser’s Theorem was later shown to hold under the weaker hypothesis n≥ d∗(G) , where 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