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A classical tool in the study of real closed fields are the fields $K((G))$ of generalised power series (i.e., formal sums with well-ordered support) with coefficients in a field $K$ of characteristic 0 and exponents in an ordered abelian…

The von Weizs\"{a}cker theorem states that every sequence of nonnegative random variables has a subsequence which is Ces\`{a}ro convergent to a nonnegative random variable which might be infinite. The goal of this note is to provide a…

Kucera and Gacs independently showed that every infinite sequence is Turing reducible to a Martin-Lof random sequence. This result is extended by showing that every infinite sequence S is Turing reducible to a Martin-Lof random sequence R…

For every $\beta\in(0,\infty)$, $\beta\neq 1$ we prove that a positive measure subset $A$ of the unit square contains a point $(x_0,y_0)$ such that $A$ nontrivially intersects curves $y-y_0 = a (x-x_0)^\beta$ for a whole interval…

Every countable topological group $G$ has a closed discrete subset $A$ such that $G=AA^{-1}.$

The central result of this paper is the small-is-very-small principle for restricted sequential theories. The principle says roughly that whenever the given theory shows that a property has a small witness, i.e. a witness in every definable…

We address two questions of Simon Thomas. First, we show that for any n>2 one can find a four generated free subgroup of SLn(Z) which is profinitely dense. More generally, we show that an arithmetic group \Gamma which admits the congruence…

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 $B$…

The usual way to investigate the statistical properties of finitely generated subgroups of free groups, and of finite presentations of groups, is based on the so-called word-based distribution: subgroups are generated (finite presentations…

Let $X$ be a compact metrizable abelian group and $\mathbf{u}=\{u_n\}$ be a sequence in its dual $X^{\wedge}$. Set $s_{\mathbf{u}} (X)= \{x: (u_n,x)\to 1\}$ and $\mathbb{T}_0^H = \{(z_n)\in \mathbb{T}^{\infty} : z_n\to 1 \}$. Let $G$ be a…

A general sieve method for groups is formulated. It enables one to "measure" subsets of a finitely generated group. As an application we show that if $\Gamma$ is a finitely generated non virtually-solvable linear group of characteristic…

We consider sequences of finitely generated discrete subgroups Gamma_i=rho_i(Gamma) of a rank 1 Lie group G, where the representations rho_i are not necessarily faithful. We show that, for algebraically convergent sequences (Gamma_i),…

Let $G\cong \mathbb Z/m_1\mathbb Z\times\ldots\times \mathbb Z/m_r\mathbb Z$ be a finite abelian group with $m_1\mid\ldots\mid m_r=\exp(G)$. The Kemperman Structure Theorem characterizes all subsets $A,\,B\subseteq G$ satisfying…

Let $G$ be a finite group and construct a graph $\Delta(G)$ by taking $G\setminus\{1\}$ as the vertex set of $\Delta(G)$ and by drawing an edge between two vertices $x$ and $y$ if $\langle x,y\rangle$ is cyclic. Let $K(G)$ be the set…

We associate with every etale groupoid G two normal subgroups S(G) and A(G) of the topological full group of G, which are analogs of the symmetric and alternating groups. We prove that if G is a minimal groupoid of germs (e.g., of a group…

Let G be a finite additive abelian group with exponent exp(G)=n>1 and let A be a nonempty subset of {1,...,n-1}. In this paper, we investigate the smallest positive integer $m$, denoted by s_A(G), such that any sequence {c_i}_{i=1}^m with…

We prove that, for a free action $\alpha \colon G \curvearrowright X$ of a countably infinite discrete amenable group on a compact metric space, the small boundary property is implied by uniform property $\Gamma$ of the Cartan subalgebra…

We show that finitely generated irreducible $\mathrm{II}_1$ subfactors are generic in the following sense. Given a separable $\mathrm{II}_1$ factor $M$ and an integer $n\geq 2$, equip the set of $n$-tuples of self-adjoint operators in $M$…

We endow a topological group $(G, \tau)$ with a coarse structure defined by the smallest group ideal $S_{\tau} $ on $G$ containing all converging sequences with their limits and denote the obtained coarse group by $(G, S_{\tau})$. If $G$ is…

We prove finiteness properties for groups of homeomorphisms that have finitely many "singular points", and we describe the normal structure of such groups. As an application, we prove that every countable abelian group can be embedded into…