arXiv subjects

Aleksander Cieślak

Publications and source records attributed to Aleksander Cieślak.

7 recordsLinked to original sources

arXiv2026

Combinatorics of translations of meager and closed measure zero sets

We study combinatorial properties of translations of meager and closed sets of measure zero. We discuss constellations of Borel conjectures for related classes of small sets and show that there are no uncountable null-additive sets in the Miller model. We also study cardinal invariants of those classes. In particular, we investigate the cardinal invariants of $σ$-ideals $\mathcal{H}_{F}$ related to meager-additive sets. We obtain a new characterization of additivity of meager ideal and answer some questions of Cardona. We also show that $\mathrm{non}(\mathcal{E}^{*})$, that is equal to the translation version of the covering number of the ideal $\mathcal{E}$, is close to the cardinal invariant related to the Laver property. This strengthens a result of Bartoszyński and Judah and a result of Elekes and Steprāns. Finally, we show that every $\mathcal{E}$-Luzin set is in $\mathcal{E}^{*}$.

math.LO↗

arXiv2026

On antichain numbers and the splitting ideal

In this article, we study combinatorial properties of a certain ideal on $ω$, called the \emph{Splitting ideal}. We calculate its cardinal invariants and its position in the Katětov order among other definable ideals. We also study the antichain numbers $\mathfrak{a}(\mathcal{J})$ of algebras $\mathcal{P}(ω)/\mathcal{J}$ for various Borel ideals. We show that $\textrm{min}\{\mathfrak{b},\textrm{cov}^{+}_{h}(\mathcal{J})\}\leq\mathfrak{a}(\mathcal{J})$ holds for a wide class of ideals, including all $F_σ$-ideals, all analytic $P$-ideals and many other examples. We also show that $\mathfrak{b}\leq\mathfrak{a}(\mathcal{J})$ holds for \emph{convergent ideal} and for \emph{Boring ideal}. Finally, we will show the consistency of $\mathfrak{a}(\mathcal{J})<\mathfrak{b}$ for the \emph{Van der Waerden's ideal} and the linear growth ideal

math.LO↗

arXiv2026

On Star operation and some ideals on the Baire space

We investigate several $σ$-ideals on the Baire space $ω^ω$ $(\mathbb{Z}^ω)$, introducing and studying the ideals $\mathcal{G}$ and $\mathcal{SMZ}^+$, alongside the classical ideals of meager sets, strong measure zero sets, the eventually different ideal and infinitely equal ideal We establish structural relationships and proper inclusions among these ideals. Also we compute the cardinal invariants of $\mathcal{M}_-$, proving that they are same as invariants of $σ$-ideal of meager sets. We further analyze the operation $^*$ on families of sets, establishing dual relationships such as $\mathcal{ED}^* = \mathcal{IE}$, $\mathcal{IE}^*=\mathcal{ED}$, $\mathcal{M}_-^*=\mathcal{SMZ}^+$ and $\mathcal{H}^* = \mathcal{G}$, and derive separations between ideals under additional set-theoretic assumptions. Finally, we prove a tree dichotomy theorem for the ideal $\mathcal{M}_-$ and we study the associated forcing notion.

math.LO↗

arXiv2026

Cardinal invariants of idealized Miller null sets

This paper provides an extensive study of the $\mathscr{I}$-Miller null ideals $M_\mathscr{I}$, $σ$-ideals on the Baire space parametrized by ideals $\mathscr{I}$ on countable sets. These $σ$-ideals are associated to the idealized versions of Miller forcing in the same way that the meager ideal is associated to Cohen forcing. We compute the cardinal invariants of $M_\mathscr{I}$ for typical examples of Borel ideals $\mathscr{I}$ and show that Cichoń's Maximum can be extended by adding the uniformity and covering numbers of $M_\mathscr{I}$ for different ideals $\mathscr{I}$.

math.LO↗

arXiv2024

On ideals related to Laver and Miller trees

In this work we consider the ideals $m^0(\mathcal{I})$ and $\ell^0(\mathcal{I})$, ideals generated by the $\mathcal{I}$-positive Miller trees and $\mathcal{I}$-positive Laver trees, respectively. We investigate in which cases these ideals have cofinality larger than $\mathfrak{c}$ and we calculate some cardinal invariants closely related to these ideals.

math.LO↗

arXiv2021

Nonmeasurable images

In this article we will investigate nonmeasurability with respect to some $σ$-ideals in Polish space $X,$ of images of subsets of $X$ by selected mappings defined on the space $X$. Among of them we answer the following question: "It is true that there exists a subset of the unit disc in the real plane such that the continuum many projections onto lines are Lebesgue measurable and continuum many projections are not?". It is known that there exists continuous function $f:[0,1]\to [0,1]$ such that for every Bernstein set $B\subseteq [0,1]$ we have $f[B]=[0,1].$ We show relative consistency with $ZFC$ of fact that the above result is not true for some $\cn$ or $\cm$-completely nonmeasurable sets, even if we take less than $\c$ many continuous functions.

math.GN↗

arXiv2019

Universal sets for ideals

In this paper we consider a notion of universal sets for ideals. We show that there exist universal sets of minimal Borel complexity for classic ideals like null subsets of $2^ω$ and meager subsets of any Polish space, and demonstrate that the existence of such sets is helpful in establishing some facts about the real line in generic extensions. We also construct universal sets for $\mathcal{E}$ - the $σ$-ideal generated by closed null subsets of $2^ω$, and for some ideals connected with forcing notions: $\mathcal{K}_σ$ subsets of $ω^ω$ and the Laver ideal. We also consider Fubini products of ideals and show that there are $Σ^0_3$ universal sets for $\mathcal{N}\otimes\mathcal{M}$ and $\mathcal{M}\otimes\mathcal{N}$.

math.GN↗