相关论文: Freiman's Theorem in an arbitrary abelian group

In this paper, we prove a series of results on group embeddings in groups with a small number of generators. We show that each finitely generated group $G$ lying in a variety ${\mathcal M}$ can be embedded in a $4$-generated group $H \in…

A characterization of the finite-dimensional Leibniz algebras with an abelian subalgebra of codimension two over a field $\mathbb{F}$ of characteristic $p\neq2$ is given. In short, a finite-dimensional Leibniz algebra of dimension $n$ with…

We show, under some natural conditions, that the set of abelian points on the non-anomalous subset of a closed irreducible subvariety $X$ intersected with the union of connected algebraic subgroups of codimension at least $\dim X$ in a…

We prove a conjecture of Helfgott on the structure of sets of bounded tripling in bounded rank, which states the following. Let $A$ be a finite symmetric subset of $\mathrm{GL}_n(\mathbf{F})$ for any field $\mathbf{F}$ such that $|A^3| \leq…

There has been considerable interest in recent decades in questions of random generation of finite and profinite groups, and finite simple groups in particular. In this paper we study similar notions for finite and profinite associative…

Let $A$ be a subset of $G$, where $G$ is a finite abelian group of torsion $r$. It was conjectured by Ruzsa that if $|A+A|\leq K|A|$, then $A$ is contained in a coset of $G$ of size at most $r^{CK}|A|$ for some constant $C$. The case $r=2$…

We give an analog of Frobenius' theorem about the factorization of the group determinant on the group algebra of finite abelian groups and we extend it into dihedral groups and generalized quaternion groups. Furthermore, we describe the…

For every finite abelian group $G$, there are positive integers $n$ and $d$ such that $G$ is isomorphic to the multiplicative group of $d$-th powers of reduced residues modulo $n$.

We prove:(1) the existence, for every integer n > 3, of a noncompact smooth n-dimensional topological manifold whose diffeomorphism group contains an isomorphic copy of every finitely presented group; (2) a finiteness theorem on finite…

We study finitely generated groups whose word problems are accepted by counter automata. We show that a group has word problem accepted by a blind n-counter automaton in the sense of Greibach if and only if it is virtually free abelian of…

We extend Ueda's peak set theorem for subdiagonal subalgebras of tracial finite von Neumann algebras, to sigma-finite von Neumann algebras (that is, von Neumann algebras with a faithful state; which includes those on a separable Hilbert…

Let K be a number field and let S be a finite set of places of K which contains all the Archimedean places. For any f(z) in K(z) of degree d at least 2 which is not a d-th power in \bar{K}(z), Siegel's theorem implies that the image set…

Let $(G,+)$ be a countable abelian group such that the subgroup $\{g+g\colon g\in G\}$ has finite index and the doubling map $g\mapsto g+g$ has finite kernel. We establish lower bounds on the upper density of a set $A\subset G$ with respect…

We prove that if a set is `large' in the sense of Erd\H{o}s, then it approximates arbitrarily long arithmetic progressions in a strong quantitative sense. More specifically, expressing the error in the approximation in terms of the gap…

Federer's characterization of sets of finite perimeter states (in Euclidean spaces) that a set is of finite perimeter if and only if the measure-theoretic boundary of the set has finite Hausdorff measure of codimension one. In complete…

The complete affine structures on abelian Lie algebras in small dimensions are well known. In this paper we are interested by the non complete case. In particular we classify all these structures in dimensions 2 and 3.

For a finite multiset $A$ of an abelian group $G$, let $\text{FS}(A)$ denote the multiset of the $2^{|A|}$ subset sums of $A$. It is natural to ask to what extent $A$ can be reconstructed from $\text{FS}(A)$. We fully solve this problem for…

It is proved that the assembly maps in algebraic K- and L-theory with respect to the family of finite subgroups is injective for groups with finite asymptotic dimension that admit a finite model for the classifying space for proper actions.…

We characterize the pairs of sets $A, B$ in an arbitrary (countable or uncountable) discrete abelian group $\Gamma$ satisfying $\tilde{m}(A+B)<\tilde{m}(A)+\tilde{m}(B)$, where $\tilde{m}$ is an arbitrary finitely additive…

A group is small if it has countably many complete $n$-types over the empty set for each natural number n. More generally, a group $G$ is weakly small if it has countably many complete 1-types over every finite subset of G. We show here…