Related papers: Borel sets without perfectly many overlapping tran…

For a cardinal lambda<lambda_{omega_1} we give a ccc forcing notion P which forces that for some Borel subset B of the Cantor space (1) there a sequence (eta_alpha:alpha<lambda) of distinct elements such that |(eta_alpha+B) cap…

For a countable ordinal epsilon we construct a Sigma^0_2 subset of the Cantor space for which one may force aleph_epsilon translations with intersections of size 2i, but such that it has no perfect set of such translations in any ccc…

We show that, consistently, there is a Borel set which has uncountably many pairwise very non-disjoint translations, but does not allow a perfect set of such translations.

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…

We give an affirmative answer to the following question: Is any Borel subset of a Cantor set $\textbf{ C}$ a sum of a countable number of pairwise disjoint $h$-homogeneous subspaces that are closed in $X$? It follows that every Borel set $X…

We study the Borel and analytic subsets of the spaces \({}^{\kappa}\kappa\) and \({}^{\kappa}2\) endowed with ideal topologies, where \(\kappa\) is a regular uncountable cardinal. We establish that the Borel hierarchy does not collapse in…

A Cantor set is a non-empty, compact set that has neither interior nor isolated points. In this paper a Cantor set $K\subseteq \mathbb{R}$ is constructed such that every set definable in $(\mathbb{R},<,+,\cdot,K)$ is Borel. In addition, we…

For $\lambda\in(0,1/3]$ let $C_\lambda$ be the middle-$(1-2\lambda)$ Cantor set in $\mathbb R$. Given $t\in[-1,1]$, excluding the trivial case we show that \[ \Lambda(t):=\left\{\lambda\in(0,1/3]:…

We answer a question of Darji and Keleti by proving in $ZFC$ that there exists a compact nullset $C_0\subset\RR$ such that for every perfect set $P\subset\RR$ there exists $x\in\RR$ such that $(C_0+x)\cap P$ is uncountable. Using this $C_0$…

Generalizing classical descriptive set theory opens foundational questions about the Borel hierarchy. In this paper we systematically study those questions, working in the general framework of Polish-like spaces relative to an uncountable…

In this paper we prove three theorems about the theory of Borel sets in models of ZF without any form of the axiom of choice. We prove that if B is a G-delta-sigma set, then either B is countable or B contains a perfect subset. Second, we…

We show that for a $\sigma $-ideal $\ci$ with a Borel base of subsets of an uncountable Polish space, if $\ca$ is (in several senses) a "regular" family of subsets from $\ci $ then there is a subfamily of $\ca$ whose union is completely…

Given $\lambda\in(0, 1)$, let $E_\lambda$ be the self-similar set generated by the iterated function system $\{x/3,(x+\lambda)/3,(x+2)/3\}$. Then $E_\lambda$ is a self-similar set with overlaps. We obtain the necessary and sufficient…

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…

There exists a family $\{B_{\alpha}\}_{\alpha<\omega_1}$ of sets of countable ordinals such that o $\max B_{\alpha}=\alpha$, o if $\alpha\in B_{\beta}$ then $B_{\alpha}\subseteq B_{\beta}$, o if $\lambda\leq \alpha$ and $\lambda$ is a limit…

In relation to the Erd\H os similarity problem (show that for any infinite set $A$ of real numbers there exists a set of positive Lebesgue measure which contains no affine copy of $A$) we give some new examples of infinite sets which are…

Assume that there is no quasi-measurable cardinal smaller than $2^\omega$. ($\kappa$ is quasi measurable if there exists $\kappa $-additive ideal $\ci $ of subsets of $\kappa $ such that the Boolean algebra $P(\kappa)/\ci$ satisfies c.c.c.)…

Let $0 < \beta \leq \alpha < 1$ and $\kappa > 0$. I prove that there exists $\eta > 0$ such that the following holds for every pair of Borel sets $A,B \subset \mathbb{R}$ with $\dim_{\mathrm{H}} A = \alpha$ and $\dim_{\mathrm{H}} B =…

We investigate some self-similar Cantor sets $C(l,r,p)$, which we call S-Cantor sets, generated by numbers $l,r,p \in \mathbb{N}$, $l+r<p$. We give a full characterization of the set $C(l_1,r_1,p)-C(l_2,r_2,p)$ which can take one of the…

We answer a question of Darji and Keleti by proving that there exists a compact set $C_0\subset\RR$ of measure zero such that for every perfect set $P\subset\RR$ there exists $x\in\RR$ such that $(C_0+x)\cap P$ is uncountable. Using this…