Related papers: Naively Haar null sets in Polish groups

In this paper we discuss the problem of existence of so called weak Sierpi\'nski sets in groups. It is known that group $G$ has a Sierpi\'nski subset if and only if it contains a free subgroup. In their paper, Tomkowicz and Wagon…

We prove that there exists a countable metrizable topological group $G$ such that every countable metrizable group is isomorphic to a quotient of $G$. The completion $H$ of $G$ is a Polish group such that every Polish group is isomorphic to…

The notion of a shift-compact set in an abelian topological group $X$ plays a significant role in functional equations and inequalities, especially so since each Borel set that is not Haar-meagre, alternatively not Haar-null, is necessarily…

In this paper, we prove several results concerning Polish group topologies on groups of non-singular transformation. We first prove that the group of measure-preserving transformations of the real line whose support has finite measure…

Letting A be a Borel subset of n dimensional Euclidean space, and W(x) be an m dimensional affine subspace containing x and varying in a Lipschitz way according to x, we establish that A is Lebesgue null if and only if $A \cap W(x)$ has m…

We study several notions of null sets on infinite-dimensional Carnot groups. We prove that a set is Aronszajn null if and only if it is null with respect to measures that are convolutions of absolutely continuous (CAC) measures on Carnot…

We expand the results of Roslanowski and Shelah arXive:1806.06283 , arXive:1909.00937 to all perfect Abelian Polish groups $(H,+)$. In particular, we show that if $\alpha<\omega_1$ and $4\leq k<\omega$, then there is a ccc forcing notion…

In this article, we introduce a formal definition of the concept of probability tree and conduct a detailed and comprehensive study of its fundamental structural properties. In particular, we define what we term an inductive probability…

Let $G$ be an abelian group of finite order $n$, and let $h$ be a positive integer. A subset $A$ of $G$ is called {\em weakly $h$-incomplete}, if not every element of $G$ can be written as the sum of $h$ distinct elements of $A$; in…

If $(X,d)$ is a Polish metric space of dimension $0$, then by Wadge's lemma, no more than two Borel subsets of $X$ can be incomparable with respect to continuous reducibility. In contrast, our main result shows that for any metric space…

In this paper we further develop the theory of canonical approximations of Polishable subgroups of Polish groups, building on previous work of Solecki and Farah--Solecki. In particular, we obtain a characterization of such canonical…

We study classes of Borel subsets of the real line $\mathbb{R}$ such as levels of the Borel hierarchy and the class of sets that are reducible to the set $\mathbb{Q}$ of rationals, endowed with the Wadge quasi-order of reducibility with…

We show that under some conditions on a family $\mathcal{A}\subset\bbi$ there exists a subfamily $\mathcal{A}_0\subset\mathcal{A}$ such that $\bigcup \mathcal{A}_0$ is nonmeasurable with respect to a fixed ideal $\bbi$ with Borel base of a…

A topological group G is profinite if it is compact and totally disconnected. Equivalently, G is the inverse limit of a surjective system of finite groups carrying the discrete topology. We discuss how to represent a countably based…

A Borel probability measure $\mu$ on a locally compact group is called a spectral measure if there exists a subset of continuous group characters which forms an orthogonal basis of the Hilbert space $L^2(\mu)$. In this paper, we…

We show that a compact group $G$ has finite conjugacy classes, i.e., is an FC-group if and only if its center $Z(G)$ is open if and only if its commutator subgroup $G'$ is finite. Let $d(G)$ denote the Haar measure of the set of all pairs…

Let G be a semisimple linear algebraic group defined over rational numbers, K be a maximal compact subgroup of its real points and {\Gamma} be an arithmetic lattice. One can associate a probability measure {\mu}(H) on {\Gamma}\G for each…

This paper studies the structure of the Mordell--Weil groups of semiabelian varieties over algebraic extensions of number fields whose absolute Galois group is finitely generated, with particular emphasis on that generated by a single…

Let $k$ be any positive integer and $G$ a compact (Hausdorff) group. Let $\mf{np}_k(G)$ denote the probability that $k+1$ randomly chosen elements $x_1,\dots,x_{k+1}$ satisfy $[x_1,x_2,\dots,x_{k+1}]=1$. We study the following problem: If…

Let $G$ be a simple algebraic group over an algebraically closed field and let $X$ be an irreducible subvariety of $G^r$ with $r \geqslant 2$. In this paper, we consider the general problem of determining if there exists a tuple $(x_1,…