Related papers: Sets with more sums than differences

Let $A$ be a finite, nonempty subset of an abelian group. We show that if every element of $A$ is a sum of two other elements, then $A$ has a nonempty zero-sum subset. That is, a (finite, nonempty) sum-full subset of an abelian group is not…

We discuss several questions concerning sum-free sets in groups, raised by Erd\H{o}s in his survey "Extremal problems in number theory" (Proceedings of the Symp. Pure Math. VIII AMS) published in 1965. Among other things, we give a…

We show that there exist infinite sets $A = \{a_1,a_2,\dots\}$ and $B = \{b_1,b_2,\dots\}$ of natural numbers such that $a_i+b_j$ is prime whenever $1 \leq i < j$.

In this paper, a new data structure named group-list is proposed. The group-list is as simple as the inverted index. However, the group-list divides document identifiers in an inverted index into groups, which makes it more efficient when…

A Sidon set is a set A of integers such that no integer has two essentially distinct representations as the sum of two elements of A. More generally, for every positive integer g, a B_2[g]-set is a set A of integers such that no integer has…

Problems in additive number theory related to sum and difference sets, more general binary linear forms, and representation functions of additive bases for the integers and nonnegative integers.

We show that if $A=\{a_1 < a_2 < \ldots < a_k\}$ is a set of real numbers such that the differences of the consecutive elements are distinct, then for and finite $B \subset \mathbb{R}$, $$|A+B|\gg |A|^{1/2}|B|.$$ The bound is tight up to…

A set $A$ is an $(r,\ell)$-approximate group in the additive abelian group $G$ if $A$ is a nonempty subset of $G$ and there exists a subset $X$ of $G$ such that $|X| \leq \ell$ and $rA \subseteq X+A$. The set $A$ is an asymptotic…

Let $X$ be an $n$-element set. A set-pair system $\mbox{$\cal P$}=\{(A_i,B_i)\}_{1\leq i\leq m}$ is a collection of pairs of disjoint subsets of $X$. It is called skew Bollob\'as system if $A_i\cap B_j\neq \emptyset$ for all $1\leq i<j \leq…

A totally symmetric set is a finite subset of a group for which any permutation of the elements can be realized by conjugation in the ambient group. Such sets are rigid under homomorphisms, and so exert a great deal of control over the…

Let D\subseteq \mathbb{R} be closed and discrete and f:D^n \to \mathbb{R} be such that f(D^n) is somewhere dense. We show that (\mathbb{R},+,\cdot,f) defines the set of integers. As an application, we get that for every a,b \in \mathbb{R}…

Let F be a finite nonempty family of finite nonempty sets. We prove the following: (i) F satisfies the condition of the title if and only if for every pair of distinct subfamilies {A_1,...,A_r}, {B_1,...,B_s} of F, the union of the A_i is…

Let $a(n)$ denote the number of non-isomorphic abelian groups with $n$ elements. In 1991 Ivi\'{c} proved an asymptotic formula of the sum $\sum_{n\leq x} a(n+ a(n)).$ In this paper, we will prove a sharper asymptotic formula for this sum.

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 $h$…

Cantor sets of integers have a rich set of arithmetic combinatorial properties. We consider classical Cantor sets, with a base and a fixed set of allowed digits. For such sets, we (a) give examples of such sets that satisfy the intersective…

We prove some new bounds for the size of the maximal dissociated subset of structured (having small sumset, large energy and so on) subsets A of an abelian group.

We study the structure of infinite discrete sets D definable in expansions of ordered Abelian groups whose theories are strong and definably complete, with particular emphasis on the set D' comprised of differences between successive…

We develop a new method leading the structure of finite subsets S and T of an abelian group with $|S+T|\le |S|+|T|$. We show also how to recover the known results in this area in a relatively short space.

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…

Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-intersecting if each member of $\mathcal{A}$ intersects each member of $\mathcal{B}$. For any two integers $n$ and $k$ with $0 \leq k \leq n$, let ${[n] \choose \leq…