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 \(\Sigma _n(S)\) 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, \(|\Sigma _n(S)|\le |S|-n+1\), it is known that the terms of S can be partitioned into n nonempty sets \(A_1,\ldots ,A_n\subseteq G\) such that \(\Sigma _n(S)=A_1+\ldots +A_n\). Moreover, if the upper bound is strict, then \(|A_i\setminus Z|\le 1\) for all i, where \(Z=\bigcap _{i=1}^{n}(A_i+H)\) and \(H=\{g\in G:\; g+\Sigma _n(S)=\Sigma _n(S)\}\) is the stabilizer of \(\Sigma _n(S)\). This allows structural results for sumsets to be used to study the subsum set \(\Sigma _n(S)\) 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 \(A_i\) of as near equal a size as possible, so \(\lfloor \frac{|S|}{n}\rfloor \le |A_i|\le \lceil \frac{|S|}{n}\rceil \) for all i, apart from one highly structured counterexample when \(|\Sigma _n(S)|= |S|-n+1\) with \(n=2\). The added information of knowing the sets \(A_i\) are of near equal size can be of use when applying the aforementioned partitioning result, or when applying sumset results to study \(\Sigma _n(S)\) (e.g., [20]). We also give an extension increasing the flexibility of the aforementioned partitioning result and prove some stronger results when \(n\ge \frac{1}{2}|S|\) is very large.