Representing Sequence Subsums as Sumsets of Near Equal Sized Sets

Abstract

For a sequence S of terms from an abelian group G of length |S|, let Σn(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, |Σn(S)| ≤ |S| − n + 1, it is known that the terms of S can be partitioned into n nonempty sets A1, . . . , An ⊆ G such that Σn(S) = A1 + . . . + An. Moreover, if the upper bound is strict, then |Ai \ Z| ≤ 1 for all i, where Z = n i=1 (Ai + H) and H = {g ∈ G : g + Σn(S) = Σn(S)} is the stabilizer of Σn(S). This allows structural results for sumsets to be used to study the subsum set Σn(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 |Σn(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 Σn(S) (e.g., [20]). We also give an extension increasing the flexibility of the aforementioned partitioning result and prove some stronger results when n ≥ 1 2 |S| is very large.