Related papers: Effective results on the size and structure of sum…

Let $A \subset \mathbb{Z}^d$ be a finite set. It is known that the sumset $NA$ has predictable size ($\vert NA\vert = P_A(N)$ for some $P_A(X) \in \mathbb{Q}[X]$) and structure (all of the lattice points in some finite cone other than all…

Let $A = \{0 = a_0 < a_1 < \cdots < a_{\ell + 1} = b\}$ be a finite set of non-negative integers. We prove that the sumset $NA$ has a certain easily-described structure, provided that $N \geqslant b-\ell$, as recently conjectured by Shakan…

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…

Let $A\subseteq \mathbb{Z}_{\geq 0}$ be a finite set with minimum element $0$, maximum element $m$, and $\ell$ elements strictly in between. Write $(hA)^{(t)}$ for the set of integers that can be written in at least $t$ ways as a sum of $h$…

A given subset $A$ of natural numbers is said to be complete if every element of $\N$ is the sum of distinct terms taken from $A$. This topic is strongly connected to the knapsack problem which is known to be NP complete. The main goal of…

Sumsets are central objects in additive combinatorics. In 2007, Granville asked whether one can efficiently recognize whether a given set $S$ is a sumset, i.e. whether there is a set $A$ such that $A+A=S$. Granville suggested an algorithm…

Let X be a normed space. A subset A of X is approximately convex if $d(ta+(1-t)b,A) \le 1$ for all $a,b \in A$ and $t \in [0,1]$ where $d(x,A)$ is the distance of $x$ to $A$. Let $\Co(A)$ be the convex hull and $\diam(A)$ the diameter of…

We investigate additive properties of sets $A,$ where $A=\{a_1,a_2,\ldots ,a_k\}$ is a monotone increasing set of real numbers, and the differences of consecutive elements are all distinct. It is known that $|A+B|\geq c|A||B|^{1/2}$ for any…

Given a measurable set $A\subset \R^n$ of positive measure, it is not difficult to show that $|A+A|=|2A|$ if and only if $A$ is equal to its convex hull minus a set of measure zero. We investigate the stability of this statement: If…

Merging together a result of Nathanson from the early 70s and a recent result of Granville and Walker, we show that for any finite set $A$ of integers with $\min(A)=0$ and $\gcd(A)=1$ there exist two sets, the "head" and the "tail", such…

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…

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…

In this paper we give a different approach to determining the cardinality of $h$-fold sumsets $hA$ when $A\subset \mathbb{Z}^d$ has $d+2$ elements. This enables us to provide more general result with a shorter and simpler proof. We also…

Given a large finite point set, $P\subset \mathbb R^2$, we obtain upper bounds on the number of triples of points that determine a given pair of dot products. That is, for any pair of positive real numbers, $(\alpha, \beta)$, we bound the…

Steinitz's theorem states that if a point $a \in \mathrm{int\,conv\,} X$ for a set $X \subset \mathbb{R}^d$, then $X$ contains a subset $Y$ of size at most $2d$ such that $a \in \mathrm{int\,conv\,}Y$. The bound $2d$ is best possible here.…

Given a finite set of lattice points, we compare its sumsets and lattice points in its dilated convex hulls. Both of these are known to grow as polynomials. Generally, the former are subsets of the latter. In this paper, we will see that…

We study a combinatorial notion where given a set of lattice points one takes the set of all sums of subsets of a fixed size, and we ask if the given set comes from a convex lattice polytope whether the resulting set also comes from a…

We give an explicit upper bound on the volume of lattice simplices with fixed positive number of interior lattice points. The bound differs from the conjectural sharp upper bound only by a linear factor in the dimension. This improves…

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 construct finite sets of real numbers that have a small difference set and strong local properties. In particular, we construct a set $A$ of $n$ real numbers such that $|A-A|=n^{\log_2 3}$ and that every subset $A'\subseteq A$ of size…