Representing Sequence Subsums as Sumsets of Near Equal Sized Sets


For a sequence S of terms from an abelian group G of length |S|, let Σ n(S) denote the set of all elements that can be represented as the sum of terms in some n-term subsequence of S. When the subsum set is very small, | Σ n(S) | ≤ | S| - n+ 1, it is known that the terms of S can be partitioned into n nonempty sets A1, …, An⊆ G such that Σ n(S) = A1+ … + An. Moreover, if the upper bound is strict, then | Ai\ Z| ≤ 1 for all i, where Z=⋂i=1n(Ai+H) and H={g∈G:g+Σn(S)=Σn(S)} is the stabilizer of Σ n(S). This allows structural results for sumsets to be used to study the subsum set Σ n(S) and is one of the two main ways to derive the natural subsum analog of Kneser’s Theorem for sumsets. In this paper, we show that such a partitioning can be achieved with sets Ai of as near equal a size as possible, so ⌊|S|n⌋≤|Ai|≤⌈|S|n⌉ for all i, apart from one highly structured counterexample when | Σ n(S) | = | S| - n+ 1 with n= 2. The added information of knowing the sets Ai are of near equal size can be of use when applying the aforementioned partitioning result, or when applying sumset results to study Σ n(S) (e.g., [20]). We also give an extension increasing the flexibility of the aforementioned partitioning result and prove some stronger results when n≥12|S| is very large.

Publication Title

Springer Proceedings in Mathematics and Statistics