Related papers: Sum-full sets are not zero-sum-free

Let $G$ be a finite group and $\Z G$ its integral group ring. We show that if $\alpha$ is a non-trivial bicyclic unit of $\Z G$, then there are bicyclic units $\beta$ and $\gamma$ of different types, such that $\GEN{\alpha,\beta}$ and…

A corollary of Kneser's theorem, one sees that any finite non-empty subset $A$ of an abelian group $G = (G,+)$ with $|A + A| \leq (2-\eps) |A|$ can be covered by at most $\frac{2}{\eps}-1$ translates of a finite group $H$ of cardinality at…

It was shown in Part I that there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose subgroups are either cyclic or Zariski-dense. Here we show…

In this note some properties of the sum of element orders of a finite abelian group are studied.

We define a notion of an arithmetic set in an arbitrary countable group and study properties of these sets in the cases of Abelian groups and non-abelian free groups.

We prove that the abstract commensurator of a nonabelian free group, an infinite surface group, or more generally of a group that splits appropriately over a cyclic subgroup, is not finitely generated. This applies in particular to all…

We construct a non-free but aleph_1-separable, torsion-free abelian group G with a pure free subgroup B such that all subgroups of G disjoint from B are free and such that G/B is divisible. This answers a question of Irwin and shows that a…

Let $\mathcal{A}$ be a finite set of integers, and let $h\mathcal{A}$ denote the $h$-fold sumset of $\mathcal{A}$. Let $(h\mathcal{A})^{(t)}$ be subset of $h\mathcal{A}$ consisting of all integers that have at least $t$ representations as a…

Let G be a torsion-free abelian group of finite rank. The orbits of the action of Aut(G) on the set of maximal independent subsets of G determine the indecomposable decompositions of G. G contains a direct sum of pure strongly…

Cameron and Erd\H{o}s asked whether the number of \emph{maximal} sum-free sets in $\{1, \dots , n\}$ is much smaller than the number of sum-free sets. In the same paper they gave a lower bound of $2^{\lfloor n/4 \rfloor }$ for the number of…

If $M$ is a set of nonsingular $k\times k$ matrices then for many pairs of matrices, $A,B\in M,$ the sum is nonsingular, $\det(A+B)\neq 0.$ We prove a more general statement on nonsingular sums with an application.

We give two examples of a finitely generated subgroup of a free group and a subset, closed in the profinite topology of a free group, such that their product is not closed in the profinite topology of a free group.

We study sum-free sets in sparse random subsets of even order abelian groups. In particular, we determine the sharp threshold for the following property: the largest such set is contained in some maximum-size sum-free subset of the group.…

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 every infinite, discrete abelian group admits a pair of $I_0$ sets whose union is not $I_0$. In particular, this implies that every such group contains a Sidon set that is not $I_{0}$.

Suppose that G is a nontrivial torsion-free group and w is a word over the alphabet G\cup\{x_1^{\pm1},...,x_n^{\pm1}\}. It is proved that for n\ge2 the group \~G=<G,x_1,x_2,...,x_n | w=1> always contains a nonabelian free subgroup. For n=1…

We will show that every element of a finitely generated abelian group is automorphically equivalent what we will define to be a {\em representative element} in a {\em repeat-free subgroup}, and for finite abelian groups we can count the…

In this paper, poor abelian groups are characterized. It is proved that an abelian group is poor if and only if its torsion part contains a direct summand isomorphic to $\oplus_{p \in P} \Z_p$, where $P$ is the set of prime integers. We…

It is a simple fact that a subgroup generated by a subset $A$ of an abelian group is the direct sum of the cyclic groups $\langle a\rangle$, $a\in A$ if and only if the set $A$ is independent. In [5] the concept of an $independent$ set in…

We prove that |A^n| > c_n |A|^{[\frac{n+1}{2}]} for any finite subset A of a free group if A contains at least two noncommuting elements, where c_n>0 are constants not depending on A. Simple examples show that the order of these estimates…