Related papers: Meager-additive sets in topological groups

We show that an ideal $\mathcal{I}$ on the positive integers is meager if and only if there exists a bounded nonconvergent real sequence $x$ such that the set of subsequences [resp. permutations] of $x$ which preserve the set of…

Consider a topological dynamical system where the group is abelian and the topologies are locally compact and second-countable. Given an invariant measure for this system, we show that if its dynamical spectrum is contained in some Borel…

In this work, we show that if $f$ is a uniformly continuous map defined over a Polish metric space, then the set of $f$-invariant measures with zero metric entropy is a $G_\delta$ set (in the weak topology). In particular, this set is…

We show that if $X$ is an $m$-dimensional definable set in $\mathbb{R}^\text{pow}_\text{an}$, the structure of real subanalytic sets with real power maps added, then for any positive integer r there exists a $C^r$-parameterization of X…

In a general measure space $(X,\mathcal L,\lambda)$, a characterization of weakly null sequences in $L_\infty (X,\mathcal L,\lambda)$ ($u_k \rightharpoonup 0$) in terms of their pointwise behaviour almost everywhere is derived from the…

Let $G$ be a countable residually finite group (for instance $\mathbb{F}_2$) and let $\overleftarrow{G}$ be a totally disconnected metric compactification of $G$ equipped with the action of $G$ by left multiplication. For every $r\geq 1$ we…

For a Polish group G let cov_G be the minimal number of translates of a fixed closed nowhere dense subset of G required to cover G. For many locally compact G this cardinal is known to be consistently larger than cov(meager) which is the…

We show that an ideal $\mathcal{I}$ on $\omega$ is meager if and only if the set of sequences $(x_n)$ taking values in a Polish space $X$ for which all elements of $X$ are $\mathcal{I}$-cluster points of $(x_n)$ is comeager. The latter…

We investigate fixed point properties for isometric actions of topological groups on a wide class of metric spaces, with a particular emphasis on Hilbert spaces. Instead of requiring the action to be continuous, we assume that it is…

In convex geometry, the Shapley-Folkman Lemma asserts that the nonconvexity of a Minkowski sum of $n$ dimensional bounded nonconvex sets does not accumulate once the number of summands exceeds the dimension $n$, and thus the sum becomes…

We show that the set of Liouville numbers is either null or non-$\sigma$-finite with respect to every translation invariant Borel measure on $\RR$, in particular, with respect to every Hausdorff measure $\iH^g$ with gauge function $g$. This…

We introduce the Solecki submeasure $\sigma(A)=\inf_F\sup_{x,y\in G}|F\cap xAy|/|F|$ and its left and right modifications on a group $G$, and study the interplay between the Solecki submeasure and the Haar measure on compact topological…

The concept of a C-approximable group, for a class of finite groups C, is a common generalization of the concepts of a sofic, weakly sofic, and linear sofic group. Glebsky raised the question whether all groups are approximable by finite…

We study strong types and Galois groups in model theory from a topological and descriptive-set-theoretical point of view, leaning heavily on topological dynamical tools. More precisely, we give an abstract (not model theoretic) treatment of…

A pair of non-empty subsets $(W,W')$ in an abelian group $G$ is an additive complement pair if $W+W'=G$. $W'$ is said to be minimal to $W$ if $W+(W'\setminus \{w'\}) \neq G, \forall \,w'\in W'$. In general, given an arbitrary subset in a…

All spaces are assumed to be separable and metrizable. Consider the following properties of a space $X$. (1) $X$ is Polish. (2) For every countable crowded $Q\subseteq X$ there exists a crowded $Q'\subseteq Q$ with compact closure. (3)…

Kunen's proof of the non-existence of Reinhardt cardinals opened up the research on very large cardinals, i.e., hypotheses at the limit of inconsistency. One of these large cardinals, I0, proved to have descriptive-set-theoretical…

A separable metric space X is an H-null set if any uniformly continuous image of X has Hausdorff dimension zero. upper H-null, directed P-null and P-null sets are defined likewise, with other fractal dimensions in place of Hausdorff…

If $(X,d)$ is a metric space then the map $f\colon X\to X$ is defined to be a weak contraction if $d(f(x),f(y))<d(x,y)$ for all $x,y\in X$, $x\neq y$. We determine the simplest non-closed sets $X\subseteq \mathbb{R}^n$ in the sense of…

Let $G$ be a topological group and let $\mu$ be the Lebesgue measure on the interval $[0,1]$. We let $L_0(G)$ to be the topological group of all $\mu$-equivalence classes of $\mu$-measurable functions defined on [0,1] with values in $G$,…