Related papers: Open problems about sumsets in finite abelian grou…

Let $A=\{a_{1},\ldots,a_{k}\}$ be a nonempty finite subset of an additive abelian group $G$. For a positive integer $h$, the restricted $h$-fold signed sumset of $A$, denoted by $h^{\wedge}_{\pm}A$, is defined as $$h^{\wedge}_{\pm}A =…

Let $A = \{a_1, \ldots, a_k\}$ be a nonempty finite subset of an additive abelian group $G$. For a nonnegative integer $h$, the \emph{$h$-fold signed sumset} of $A$, denoted by $h_{\pm} A$, is defined by $$ h_{\pm} A = \Biggl\{\sum_{i =…

Let $A=\{a_{1},\ldots,a_{k}\}$ be a nonempty finite subset of an additive abelian group $G$. For a positive integer $h$, the $h$-fold signed sumset of $A$, denoted by $h_{\pm}A$, is defined as $$h_{\pm}A=\left\lbrace \sum_{i=1}^{k}…

Let $G$ be an additive abelian group. Let $A=\{a_{0}, a_{1},\ldots, a_{k-1}\}$ be a nonempty finite subset of $G$. For a positive integer $h$ satisfying $1\leq h\leq k$, we let \[h\hat{}_{\underline{+}}A:=\{\Sigma_{i=0}^{k-1}\lambda_{i}…

Let $G$ be an additive abelian group and $h$ be a positive integer. For a nonempty finite subset $A=\{a_0, a_1,\ldots, a_{k-1}\}$ of $G$, we let \[h_{\underline{+}}A:=\{\Sigma_{i=0}^{k-1}\lambda_{i} a_{i}: (\lambda_{0}, \ldots,…

For a set $A$ of $k$ elements from an additive abelian group $G$ and a positive integer $r \leq k$, we consider the set of elements of $G$ that can be written as a sum of $h$ elements of $A$ with at least $r$ distinct elements. We denote…

For a finite abelian group $G$, a nonempty subset $A$ of $G$, and a positive integer $h$, we let $hA$ denote the $h$-fold sumset of $A$; that is, $hA$ is the collection of sums of $h$ not-necessarily-distinct elements of $A$. Furthermore,…

Let $A$ be a finite set of integers and let $hA$ be its $h$-fold sumset. This paper investigates the sequence of sumset sizes $( |hA| )_{h=1}^{\infty}$, the relations between these sequences for affinely inequivalent sets $A$ and $B$, and…

In this note we find the optimal lower bound for the size of the sumsets $HA$ and $H\,\hat{}A$ over finite sets $H, A$ of nonnegative integers, where $HA = \bigcup_{h\in H} hA$ and $H\,\hat{}A = \bigcup_{h\in H} h\,\hat{}A$. We also find…

Inspired by recent questions of Nathanson, we show that for any infinite abelian group $G$ and any integers $m_1, \ldots, m_H$, there exist finite subsets $A,B \subseteq G$ such that $|hA|-|hB|=m_h$ for each $1 \leq h \leq H$. We also…

For a finite abelian group $G$ and a positive integer $h$, the unrestricted (resp.~restricted) $h$-critical number $\chi(G,h)$ (resp.~$\chi \hat{\;}(G,h)$) of $G$ is defined to be the minimum value of $m$, if exists, for which the $h$-fold…

Let $A$ be a nonempty finite subset of an additive abelian group $G$. Given a nonnegative integer $h$, the $h$-fold sumset $hA$ is the set of all sums of $h$ elements of $A$, and the restricted $h$-fold sumset $h^\wedge A$ is the set of all…

For a finite abelian group $G$ and positive integers $m$ and $h$, we let $$\rho(G, m, h) = \min \{|hA| \; : \; A \subseteq G, |A|=m\}$$ and $$\rho_{\pm} (G, m, h) = \min \{|h_{\pm} A| \; : \; A \subseteq G, |A|=m\},$$ where $hA$ and…

Let $A$ be a finite set of $k$ integers. For $h \leq k$, the restricted $h$-fold sumset $h^{\wedge} A$ is the set of all sums of $h$ distinct elements of $A$. In additive combinatorics, much of the focus has traditionally been on finite…

For a finite abelian group $G$ and positive integers $m$ and $h$, we let $$\rho(G, m, h) = \min \{|hA| \; : \; A \subseteq G, |A|=m\}$$ and $$\rho_{\pm} (G, m, h) = \min \{|h_{\pm} A| \; : \; A \subseteq G, |A|=m\},$$ where $hA$ and…

This paper describes problems concerning the range of cardinalities of sumsets and restricted sumsets of finite subsets of the integers and finite subsets of ordered abelian groups.

Given a positive integer $h$ and a nonempty finite set of integers $A=\{a_{1},a_{2},\ldots,a_{k}\}$, the restricted $h$-fold signed sumset of $A$, denoted by $h^{\wedge}_{\pm}A$, is defined as $$h^{\wedge}_{\pm}A=\left\lbrace \sum_{i=1}^{k}…

Let $A$ be a set in an abelian group $G$. For integers $h,r \geq 1$ the generalized $h$-fold sumset, denoted by $h^{(r)}A$, is the set of sums of $h$ elements of $A$, where each element appears in the sum at most $r$ times. If…

It is an open problem in additive number theory to compute and understand the full range of sumset sizes of finite sets of integers, that is, the set $\mathcal{R}_{\mathbf{Z}}(h,k)= \{|hA|:A \subseteq {\mathbf{Z}} \text{ and } |A|=k\}$ for…

Let $hA$ denote the $h$-fold sumset of a subset $A$ of an abelian group. Resolving a problem of Nathanson, we show that for any prescribed permutations $\sigma_1, \ldots, \sigma_H \in \mathfrak{S}_n$, there exist finite subsets $A_1,…