Problems in additive number theory, VI: Sizes of sumsets

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 characterization for large sets $A$ in an abelian group $G$ which do not contain a subset $B$ of fixed size $k$ such that the sum of any two different elements of $B$ do not belong to $A$ (in other words, $B$ is sum-free with respect to $A$). Erd...

A set of elements of a finite abelian group is called sum-free if it contains no Schur triple, i.e., no triple of elements $x,y,z$ with $x+y=z$. The study of how large the largest sum-free subset of a given abelian group is had started more than thirty years before it was finally resolved by Green and Ruzsa a decade ago. We address the following more general question. Suppose that a set $A$ of elements of an abelian group $G$ has cardinality $a$. How many Schur triples must $...

Let A be a subset of an abelian group G. We say that A is sum-free if there do not exist x,y and z in A satisfying x + y = z. We determine, for any G, the cardinality of the largest sum-free subset of G. This equals c(G)|G| where c(G) is a constant depending on G and lying in the interval [2/7,1/2]. We also estimate the number of sum-free subsets of G. It turns out that log_2 of this number is c(G)|G| + o(|G|), which is tight up to the o-term. For certain abelian groups, ...

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 the sets $A$ with some extrema the cardinalities of $hA$.

For a subset A of a finite abelian group G we define Sigma(A)={sum_{a\in B}a:B\subset A}. In the case that Sigma(A) has trivial stabiliser, one may deduce that the size of Sigma(A) is at least quadratic in |A|; the bound |Sigma(A)|>= |A|^{2}/64 has recently been obtained by De Vos, Goddyn, Mohar and Samal. We improve this bound to the asymptotically best possible result |Sigma(A)|>= (1/4-o(1))|A|^{2}. We also study a related problem in which A is any subset of Z_{n} with al...

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 obtain some new results on $4^{\wedge}\mathcal{A}$ and $5^{\wedge}\mathcal{A}$. For example, we show that if $|\mathcal{A}|\geq 0.4045n$ and $n$ is odd, then $4^{\wedge}\mathcal{A}=\mathbb{Z}_{n}$; Under some conditions, if $n$ is even and $|...

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 $|A + A| \leq m$ is at most \[2^{o(s)}\binom{\frac{m+\beta}{2}}{s},\] where $\beta$ is the size of the largest subgroup of $G$ of size at most $\left(1+o(1)\right)m$. This bound is sharp for $\mathbb{Z}$ and many other groups. Our result imp...

We develop the theory of the additive dimension ${\rm dim} (A)$, i.e. the size of a maximal dissociated subset of a set $A$. It was shown that the additive dimension is closely connected with the growth of higher sumsets $nA$ of our set $A$. We apply this approach to demonstrate that for any small multiplicative subgroup $\Gamma$ the sequence $|n\Gamma|$ grows very fast. Also, we obtain a series of applications to the sum--product phenomenon and to the Balog--Wooley decomposi...

Let A be a finite nonempty subset of an additive abelian group G, and let \Sigma(A) denote the set of all group elements representable as a sum of some subset of A. We prove that |\Sigma(A)| >= |H| + 1/64 |A H|^2 where H is the stabilizer of \Sigma(A). Our result implies that \Sigma(A) = Z/nZ for every set A of units of Z/nZ with |A| >= 8 \sqrt{n}. This consequence was first proved by Erd\H{o}s and Heilbronn for n prime, and by Vu (with a weaker constant) for general n.

Let A be a finite subset of the integers or, more generally, of any abelian group, written additively. The set A has "more sums than differences" if |A+A|>|A-A|. A set with this property is called an MSTD set. This paper gives explicit constructions of families of MSTD sets of integers.