Related papers: Sumfree sets in groups: a survey

Denoting by Sigma(S) the set of subset sums of a subset S of a finite abelian group G, we prove that |Sigma(S)| >= |S|(|S|+2)/4-1 whenever S is symmetric, |G| is odd and Sigma(S) is aperiodic. Up to an additive constant of 2 this result is…

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 show that the optimal constant in Erd\"{o}s' sum-free subset theorem cannot be larger than $11/28 \approx .393$.

Let $A$ and $B$ be finite subsets of $\mathbb{C}$ such that $|B|=C|A|$. We show the following variant of the sum product phenomenon: If $|AB|<\alpha|A|$ and $\alpha \ll \log |A|$, then $|kA+lB|\gg |A|^k|B|^l$. This is an application of a…

It is known that $A$-paths of length $0$ mod $m$ satisfy the Erd\H{o}s-P\'osa property if $m=2$ or $m=4$, but not if $m > 4$ is composite. We show that if $p$ is prime, then $A$-paths of length $0$ mod $p$ satisfy the Erd\H{o}s-P\'osa…

Let $A$ be a subset of $\mathbb{Z} / N\mathbb{Z}$ and let $\mathcal{R}$ be the set of large Fourier coefficients of $A$. Properties of $\mathcal{R}$ have been studied in works of M.-- C. Chang, B. Green and the author. In the paper we…

In 1965, Erd\H{o}s and P\'{o}sa proved that there is an (approximate) duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold for odd cycles, and Dejter…

We address a problem posed by Nathan Kaplan in the 2014 Combinatorial and Additive Number Theory session: finding the largest subset $H \subseteq (\mathbb{Z}/4\mathbb{Z})^n$ with no distinct $x, y, z \in H$ such that $x + y + z \equiv 0…

We consider the problem of sums of dilates in groups of prime order. We show that given $A\subset \Z{p}$ of sufficiently small density then $$\big| \lambda_{1}A+\lambda_{2}A+...+ \lambda_{k}A \big|…

Let $G$ be a finite abelian group. For any positive integers $d$ and $m$, let $\varphi_G(d)$ be the number of elements in $G$ of order $d$ and $\mathsf M(G,m)$ be the set of all zero-sum sequences of length $m$. In this paper, for any…

We prove new separability results about free groups. Namely, if $H_1, \ldots , H_k$ are infinite index, finitely generated subgroups of a non-abelian free group $F$, then there exists a homomorphism onto some alternating group $f:F…

For any set $A$ of natural numbers with positive upper Banach density and any $k\geq 1$, we show the existence of an infinite set $B\subset{\mathbb N}$ and a shift $t\geq0$ such that $A-t$ contains all sums of $m$ distinct elements from $B$…

In this paper, we consider a natural generalization of the concept of order of an element in a group: an element $g \in G$ is said to have order $k$ in a subgroup $H$ of $G$ (\resp \wrt a coset $Hu$) if $k$ is the first strictly positive…

Following the sum-product paradigm, we prove that for a set $B$ with polynomial growth, the product set $B.B$ cannot contain large subsets with size of order $|B|^2$ with small doubling. It follows that the additive energy of $B.B$ is…

Let G be a finite abelian group. For g in G and i an integer we define N(i,g) to be the number of subsets of G of size i which sum up to g. We will give a short proof, using character theory, of a formula for these N(i,g) due to Li and Wan.…

A subset of the integers larger than 1 is $primitive$ if no member divides another. Erdos proved in 1935 that the sum of $1/(a\log a)$ for $a$ running over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he…

Let $\{a_1, . . . , a_n\}$ be a set of positive integers with $a_1 < \dots < a_n$ such that all $2^n$ subset sums are distinct. A famous conjecture by Erd\H{o}s states that $a_n>c\cdot 2^n$ for some constant $c$, while the best result known…

If $2 \le d \le k$ and $n \ge dk/(d-1)$, a $d$-cluster is defined to be a collection of $d$ elements of ${[n] \choose k}$ with empty intersection and union of size no more than $2k$. Mubayi conjectured that the largest size of a…

Let $A$ be a nonempty finite subset of an additive abelian group $G$. Define $A + A := \{a + b : a, b \in A\}$ and $A \dotplus A := \{a + b : a, b \in A~\text{and}~ a \neq b\}$. The set $A$ is called a {\em sum-dominant (SD) set} if $|A +…

We give a short proof of a sumset conjecture of Erd\"os, recently proved by Moreira, Richter and Robertson: every subset of the integers of positive density contains the sum of two infinite sets. The proof is written in the framework of…