Related papers: Some direct and inverse problems for the Restricte…

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 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,…

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 $A$ 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 positive integer $h$ and a subset $A$ of a given finite abelian group, we let $hA$, $h \hat{\;} A$, and $h_{\pm}A$ denote the $h$-fold sumset, restricted sumset, and signed sumset of $A$, respectively. Here we review some of what is…

Let $n$ be a positive integer, and let $A$ be a set of $k\ge 2n-1$ integers. For the restricted sumset $$ S_n(A)=\{a_1+\cdots +a_n:\ a_1,\ldots,a_n\in A,\ \text{and}\ a_i^2\neq a_j^2\ \text{for} \ 1\le i<j\le n\}, $$ by a 2002 result of Liu…

Let $h \geq 2$, $k \geq 5$ be integers and $A$ be a nonempty finite set of $k$ integers. Very recently, Tang and Xing studied extended inverse theorems for $hk-h+1 < \left|hA\right| \leq hk+2h-3$. In this paper, we extend the work of Tang…

Let $A=\{a_0,a_1,\ldots,a_{k-1}\}$ be a set of $k$ integers. For any integer $h\ge 1$ and any ordered $k$-tuple of positive integers $\mathbf{r}=(r_0,r_1,\ldots,r_{k-1})$, we define a general $h$-fold sumset, denoted by $h^{(\mathbf{r})}A$,…

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…

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…

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…

Let $A$ and $H$ be nonempty finite sets of integers and positive integers, respectively. The generalized $H$-fold sumset, denoted by $H^{(r)}A$, is the union of the sumsets $h^{(r)}A$ for $h\in H$ where, the sumset $h^{(r)}A$ is the set of…

Let $A$ be a nonempty subset of finite abelian group $G$ of order $n$. For an integer $h \geq 2$, the restricted $h$-fold sumset $h^\wedge A$ is the set of all sums of $h$ distinct elements of $A$. It is known that if $G$ is a group of…

Let $h\geq 2$ and $A=\{a_0,a_1,\ldots,a_{k-1}\}$ be a finite set of integers. It is well-known that $\left|hA\right|=hk-h+1$ if and only if $A$ is a $k$-term arithmetic progression. In this paper, we give some nontrivial inverse results of…

Let~$A$ be a set of nonnegative integers. Let~$(h A)^{(t)}$ be the set of all integers in the sumset~$hA$ that have at least~$t$ representations as a sum of~$h$ elements of~$A$. In this paper, we prove that, if~$k \geq 2$,…

Let $h\geq 2$ be a positive integer. For any subset $\mathcal{A}\subset \mathbb{Z}_n$, let $h^{\wedge}\mathcal{A}$ be the set of the elements of $\mathbb{Z}_n$ which are sums of $h$ distinct elements of $\mathcal{A}$. In this paper, we…

We consider the Exact-Weight-H problem of finding a (not necessarily induced) subgraph H of weight 0 in an edge-weighted graph G. We show that for every H, the complexity of this problem is strongly related to that of the infamous k-Sum…

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…