Related papers: Complete nonmeasurability in regular families

In the present note we study the interrelations between the sets of so-called typical numbers and numbers that are normal in base two. Employing results by Nakai and Shiokawa, we exhibit examples of numbers that belong to one set but do not…

We prove a general result on irregularities of distribution for Borel sets intersected with bounded measurable sets or affine half-spaces.

Let $X$ be a nondegenerate Peano unicoherent continuum. The family $CB(X)$ of proper subcontinua of $X$ with connected boundaries is a $G_\delta$-subset of the hyperspace $C(X)$ of all subcontinua of $X$. If every nonempty open subset of…

Let $\mu$ be a Borel measure on a compactum $X$. The main objects in this paper are $\sigma$-ideals $I(dim)$, $J_0(\mu)$, $J_f(\mu)$ of Borel sets in $X$ that can be covered by countably many compacta which are finite-dimensional, or of…

We prove that the property Add$(M)\subseteq$ Prod$(M)$ characterizes $\Sigma$-algebraically compact modules if $|M|$ is not $\omega$-measurable. Moreover, under a large cardinal assumption, we show that over any ring $R$ where $|R|$ is not…

We explore the Borel complexity of some basic families of subsets of a countable group (large, small, thin, sparse and other) defined by the size of their elements. Applying the obtained results to the Stone-\v{C}ech compactification $\beta…

The monadic theory of $(\mathbb R,\le)$ with quantification restricted to Borel sets is decidable. The Boolean combinations of $F_\sigma$ sets form an elementary substructure of the Borel sets. Under determinacy hypotheses, the proof…

We prove that the countable product of lines contains a Borel linear subspace $L\ne\mathbb R^\omega$ that cannot be covered by countably many closed Haar-meager sets. This example is applied to studying the interplay between various classes…

It is shown that there exists a normal uniform algebra, on a compact metrizable space, that fails to be strongly regular at some peak point. This answers a 31-year-old question of Joel Feinstein. Our example is R(K) for a certain compact…

We prove that it is relatively consistent with $\mathrm{ZFC}$ that every strong measure zero subset of the real line is meager-additive while there are uncountable strong measure zero sets (i.e., Borel's conjecture fails). This answers a…

We study the complexity with respect to Borel reducibility of the relations of isometry and isometric embeddability between ultrametric Polish spaces for which a set $D$ of possible distances is fixed in advance. These are, respectively, an…

The group of homeomorphisms of the closed interval that are absolutely continuous and have an absolutely continuous inverse was shown by Solecki to admit a natural Polish group topology $\tau_{ac}$. We show that, under mild conditions on a…

In this paper we address the question: How many pairwise non-isomorphic extremely amenable groups are there which are separable metrizable or even Polish? We show that there are continuum many such groups. In fact we construct continuum…

We study some strong combinatorial properties of $\textsf{MAD}$ families. An ideal $\mathcal{I}$ is Shelah-Stepr\={a}ns if for every set $X\subseteq{\left[ \omega\right]}^{<\omega}$ there is an element of $\mathcal{I}$ that either…

We introduce the notion of regular (boundary) poles for infinitesimal generators of semigroups of holomorphic self-maps of the unit disc. We characterize such regular poles in terms of $\beta$-points (i.e. pre-images of values with positive…

Any group that has a subnormal series, in which all factors are abelian and all except the last one are $p'$-torsion-free, can be embedded into a group with a subnormal series of the same length, with the same properties and such that any…

A $\sigma$-ideal $\mathcal{I}$ on a Polish group $(X,+)$ has Smital Property if for every dense set $D$ and a Borel $\mathcal{I}$-positive set $B$ the algebraic sum $D+B$ is a complement of a set from $\mathcal{I}$. We consider several…

In this paper we formulate three problems concerning topological properties of sets generating Borel non-sigma-compact groups. In case of the concrete F_\sigma\delta-subgroup of the Cantor group this gives an equivalent reformulation of the…

Following the work of Burger, Iozzi and Wienhard for representations, in this paper we introduce the notion of maximal measurable cocycles of a surface group. More precisely, let $\mathbf{G}$ be a semisimple algebraic $\mathbb{R}$-group…

We prove that topological isomorphism on procountable groups is not classifiable by countable structures, in the sense of descriptive set theory. In fact, the equivalence relation $\ell_\infty$ expressing that two sequences of reals have a…