Iterated Sumsets and Subsequence Sums

Let $G\cong \mathbb Z/m_1\mathbb Z\times\ldots\times \mathbb Z/m_r\mathbb Z$ be a finite abelian group with $m_1\mid\ldots\mid m_r=\exp(G)$. The Kemperman Structure Theorem characterizes all subsets $A,\,B\subseteq G$ satisfying $|A+B|<|A|+|B|$ and has been extended to cover the case when $|A+B|\leq |A|+|B|$. Utilizing these results, we provide a precise structural description of all finite subsets $A\subseteq G$ with $|nA|\leq (|A|+1)n-3$ when $n\geq 3$ (also when $G$ is infinite), in which case many of the pathological possibilities from the case $n=2$ vanish, particularly for large $n\geq \exp(G)-1$. The structural description is combined with other arguments to generalize a subsequence sum result of Olson asserting that a sequence $S$ of terms from $G$ having length $|S|\geq 2|G|-1$ must either have every element of $G$ representable as a sum of $|G|$-terms from $S$ or else have all but $|G/H|-2$ of its terms lying in a common $H$-coset for some $H\leq G$. We show that the much weaker hypothesis $|S|\geq |G|+\exp(G)$ suffices to obtain a nearly identical conclusion, where for the case $H$ is trivial we must allow all but $|G/H|-1$ terms of $S$ to be from the same $H$-coset. The bound on $|S|$ is improved for several classes of groups $G$, yielding optimal lower bounds for $|S|$. We also generalize Olson's result for $|G|$-term subsums to an analogous one for $n$-term subsums when $n\geq \exp(G)$, with the bound likewise improved for several special classes of groups. This improves previous generalizations of Olson's result, with the bounds for $n$ optimal.Comment: Revised version, with results reworded to appear less technica

On a Conjecture of Hamidoune for Subsequence Sums

Let G be an abelian group of order m, let S be a sequence of terms from G with k distinct terms, let m ∧ S denote the set of all elements that are a sum of some m-term subsequence of S, and let |S| be the length of S. We show that if |S| ≥ m + 1, and if the multiplicity of each term of S is at most m − k + 2, then either |m ∧ S| ≥ min{m, |S| − m + k − 1}, or there exists a proper, nontrivial subgroup Ha of index a, such that m ∧ S is a union of Ha-cosets, Ha ⊆ m ∧ S, and all but e terms of S are from the same Ha-coset, where e ≤ min{|S|−m+k−2 |Ha| − 1, a − 2} and |m ∧ S| ≥ (e + 1)|Ha|. This confirms a conjecture of Y. O. Hamidoune

Arithmetic-Progression-Weighted Subsequence Sums

Let $G$ be an abelian group, let $S$ be a sequence of terms $s_1,s_2,...,s_{n}\in G$ not all contained in a coset of a proper subgroup of $G$, and let $W$ be a sequence of $n$ consecutive integers. Let $$W\odot S=\{w_1s_1+...+w_ns_n:\;w_i {a term of} W,\, w_i\neq w_j{for} i\neq j\},$$ which is a particular kind of weighted restricted sumset. We show that $|W\odot S|\geq \min\{|G|-1,\,n\}$, that $W\odot S=G$ if $n\geq |G|+1$, and also characterize all sequences $S$ of length $|G|$ with $W\odot S\neq G$. This result then allows us to characterize when a linear equation $$a_1x_1+...+a_rx_r\equiv \alpha\mod n,$$ where $\alpha,a_1,..., a_r\in \Z$ are given, has a solution $(x_1,...,x_r)\in \Z^r$ modulo $n$ with all $x_i$ distinct modulo $n$. As a second simple corollary, we also show that there are maximal length minimal zero-sum sequences over a rank 2 finite abelian group $G\cong C_{n_1}\oplus C_{n_2}$ (where $n_1\mid n_2$ and $n_2\geq 3$) having $k$ distinct terms, for any $k\in [3,\min\{n_1+1,\,\exp(G)\}]$. Indeed, apart from a few simple restrictions, any pattern of multiplicities is realizable for such a maximal length minimal zero-sum sequence

On products of k atoms II

Let $H$ be a Krull monoid with class group $G$ such that every class contains a prime divisor (for example, rings of integers in algebraic number fields or holomorphy rings in algebraic function fields). For $k \in \mathbb N$, let $\mathcal U_k (H)$ denote the set of all $m \in \mathbb N$ with the following property: There exist atoms $u_1, ..., u_k, v_1, ..., v_m \in H$ such that $u_1 \cdot ... \cdot u_k = v_1 \cdot ...\cdot v_m$. Furthermore, let $\lambda_k (H) = \min \mathcal U_k (H)$ and $\rho_k (H) = \sup \mathcal U_k (H)$. The sets $\mathcal U_k (H) \subset \mathbb N$ are intervals which are finite if and only if $G$ is finite. Their minima $\lambda_k (H)$ can be expressed in terms of $\rho_k (H)$. The invariants $\rho_k (H)$ depend only on the class group $G$, and in the present paper they are studied with new methods from Additive Combinatorics