Related papers: The structure of higher sumsets

Given a finite set $A\subseteq \mathbb{N}$, define the sum set $$A+A = \{a_i+a_j\mid a_i,a_j\in A\}$$ and the difference set $$A-A = \{a_i-a_j\mid a_i,a_j\in A\}.$$ The set $A$ is said to be sum-dominant if $|A+A|>|A-A|$. We prove the…

A More Sums Than Differences (MSTD) set is a set $A$ for which $|A+A|>|A-A|$. Martin and O'Bryant proved that the proportion of MSTD sets in $\{0,1,\dots,n\}$ is bounded below by a positive number as $n$ goes to infinity. Iyer, Lazarev,…

A conjecture of Freiman gives an exact formula for the largest volume of a finite set $A$ of integers with given cardinality $k = |A|$ and doubling $T = |2A|$. The formula is known to hold when $T \le 3k-4$, for some small range over $3k-4$…

Let n(2,k) denote the largest integer n for which there exists a set A of k nonnegative integers such that the sumset 2A contains {0,1,2,...,n-1}. A classical problem in additive number theory is to find an upper bound for n(2,k). In this…

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,…

Let $\mathcal{A}$ be a sequence of $rk$ terms which is made up of $k$ distinct integers each appearing exactly $r$ times in $\mathcal{A}$. The sum of all terms of a subsequence of $\mathcal{A}$ is called a subsequence sum of $\mathcal{A}$.…

Fix $A$, a family of subsets of natural numbers, and let $G_A(n)$ be the maximum cardinality of a subset of $\{1,2,..., n\}$ that does not have any subset in $A$. We consider the general problem of giving upper bounds on $G_A(n)$ and give…

Let G be an additive abelian group whose finite subgroups are all cyclic. Let A_1,...,A_n (n>1) be finite subsets of G with cardinality k>0, and let b_1,...,b_n be pairwise distinct elements of G with odd order. We show that for every…

A set $A$ is MSTD (more-sum-than-difference) if $|A+A|>|A-A|$. Though MSTD sets are rare, Martin and O'Bryant proved that there exists a positive constant lower bound for the proportion of MSTD subsets of $\{1,2,\ldots ,r\}$ as…

Let $\mathcal{A}_1,\ldots,\mathcal{A}_m$ be families of $k$-subsets of an $n$-set. Suppose that one cannot choose pairwise disjoint edges from $s+1$ distinct families. Subject to this condition we investigate the maximum of…

A symmetric subset of the reals is one that remains invariant under some reflection z --> c-z. We consider, for any 0 < x <= 1, the largest real number D(x) such that every subset of $[0,1]$ with measure greater than x contains a symmetric…

This short note answers a question raised by Nathanson \cite{Nath25} about "races" between iterated sumsets. We prove that for any integer $n$, there are finite sets of integers $A$ and $B$ with same diameter such that the signs of the…

We prove that any finite abelian group $G$ contains a collection of not too many subsets with a special structure, so that for every subset $A$ of $G$ with a small doubling, there is a member $F$ of the collection that is fully contained in…

We analyze sumsets A+B = {a+b : a in A, b in B} where A,B are sets of integers, A is infinite, and B has positive upper Banach density. For each k, we show that A+B contains at least the expected density of k-term arithmetic progressions…

A finite family $\mathrsfs{F}$ of subsets of a finite set $X$ is union-closed whenever $f,g\in\mathrsfs{F}$ implies $f\cup g\in\mathrsfs{F}$. These families are well known because of Frankl's conjecture. In this paper we developed further…

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…

Sidon sets are those sets such that the sums of two of its elements never coincide. They go back to the 30s when Sidon asked for the maximal size of a subset of consecutive integers with that property. This question is now answered in a…

We investigate subsets with small sumset in arbitrary abelian groups. For an abelian group $G$ and an $n$-element subset $Y \subseteq G$ we show that if $m \ll s^2/(\log n)^2$, then the number of subsets $A \subseteq Y$ with $|A| = s$ and…

Let A={a_s(mod n_s)}_{s=0}^k be a system of residue classes. With the help of cyclotomic fields we obtain a theorem which unifies several previously known results concerning system A. In particular, we show that if every integer lies in…

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…