arXiv subjects
Publications and source records attributed to Szymon Żeberski.
Let $\mathcal{I}$ be a $σ$-ideal on the Cantor space $2^ω$. A set $X\subseteq 2^ω$ is called everywhere $\mathcal{I}$ if for any $S\in [ω]^ω$ the set $X\!\restriction\! S\in\mathcal{I}(2^S).$ We will discuss relations between such families in the class of perfect and Borel sets.
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.
The aim of this paper is to give natural examples of $\mathbfΣ_1^1$-complete and $\mathbfΠ_1^1$-complete sets. In the first part, we consider ideals on $ω$. In particular, we show that the Hindman ideal $\mathcal{H}$ is $\mathbfΠ_1^1$-complete and consider a number of ideals generated in the similar fashion. Moreover, we show that the ideal $\mathcal{D}$ is also $\mathbfΠ_1^1$-complete. In the second part, we focus on families of trees (on $ω$ and $2$) containing a specific tree type. We show the connection between two topics and explore some classical tree types (like Sacks and Miller).
This paper explores the interplay between star operations, microscopic sets, and porous sets. The study focuses on the Galvin-Mycielski-Solovay theorem, which characterizes strongly measure zero sets and their interactions with meager sets. Results include the investigation of the star operation $\mathcal{F}^*$ and its properties. The paper also examines the relationship between porous sets and microscopic sets. Additionally, the work presents constructions of families $\mathcal{F}$ in $\mathcal{P}(\mathbb{Z}), \mathcal{P}(\mathbb{Z}^ω),$ and $\mathcal{P}(2^ω)$ that satisfy $\mathcal{F} = \mathcal{F}^*$. Theorems and lemmas are provided to establish conditions under which $\mathcal{F}^{**} = \mathcal{F}$ and to analyze the implications of the Borel Conjecture and its dual. The paper concludes with a discussion of microscopic sets and their properties, including their interactions with porous sets and the non-equivalence of certain classes of sets.
In this paper, we study the translations into the Baire space of several well-known $σ$-ideals and families originally defined on the Cantor space, using their combinatorial characterizations. These include the ideals of null sets, small sets, those generated by closed measure-zero sets, and the meager sets, leading to their "fake" analogues in the Baire space. We also parametrize families related to null sets by functions from $ω^ω$. Several structural properties and relations between these families are investigated, including whether they form ideals, the existence of large chains and antichains, orthogonality, the $κ$-chain condition, and the determination of certain cardinal invariants.
We work in the Baire space $\mathbb{Z}^ω$ equipped with the coordinate-wise addition $+$. Consider a $σ-$ideal $\mathcal{I}$ and a family $\mathbb{T}$ of some kind of perfect trees. We are interested in results of the form: for every $A\in \mathcal{I}$ and a tree $T\in\mathbb{T}$ there exists $T'\in \mathbb{T}, T'\subseteq T$ such that $A+\underbrace{[T']+[T']+\dots +[T']}_{\text{n--times}}\in \mathcal{I}$ for each $n\inω$. Explored tree types include perfect trees, uniformly perfect trees, Miller trees, Laver trees and $ω-$Silver trees. The latter kind of trees is an analogue of Silver trees from the Cantor space. Besides the standard $σ$-ideal $\mathcal{M}$ of meager sets, we also analyze $\mathcal{M}_-$ and fake null sets $\mathcal{N}$. The latter two are born out of the characterizations of their respective analogues in the Cantor space. The key ingredient in proofs were combinatorial characterizations of these ideals in the Baire space.
We work in the Cantor space $2^ω$. The results of the paper adhere the following pattern. Let $\mathcal{I}\in \{\mathcal{M}, \mathcal{N}, \mathcal{M}\cap \mathcal{N}, \mathcal{E}\}$ and $T$ be a perfect, uniformly perfect or Silver tree. Then for every $A\in \mathcal{I}$ there exists $T'\subseteq T$ of the same kind as $T$ such that $A+\underbrace{[T']+[T']+\dots +[T']}_{\text{n--times}}\in \mathcal{I}$ for each $n\inω$. We also prove weaker statements for splitting trees. For the case $\mathcal{E}$ we also provide a simple characterization of basis of $\mathcal{E}$. We use these results to prove that the algebraic sum of a generalized Luzin set and a generalized Sierpiński set belongs to $u_0$ and $v_0$, provided that $\mathfrak{c}$ is a regular cardinal.
The motivation of this work are the two classical theorems on inscribing rectangles and squares into large subsets of the plane, namely Eggleston Theorem and Mycielski Theorem. Using Shoenfield Absoluteness Theorem we prove that for every Borel subset of the plane with uncountably many positive (with respect to measure or category) vertical section contains a rectangle $P\times B$ where $P$ is perfect and $B$ is Borel and positive. We also obtained a variant of Eggleston Theorem regarding the $σ$-ideal $\mathcal(E)$ generated by closed sets of measure zero. Furthermore we proved that every comeager (resp. conull) subset of the plane contains a rectangle $[T]\times H$, where $T$ is a Spinas tree containing a Silver tree and $H$ is comeager (resp. conull). Moreover we obtained a common generalization of Eggleston Theorem and Mycielski Theorem stating that every comeager (resp. conull) subset of the plane contains a rectangle $[T]\times H$ modulo diagonal, where $T$ is a uniformly perfect tree, $H$ is comeager (resp. conull) and $[T]\subseteq H$.
We study analytic and Borel subsets defined similarily to the old example of analytic complete set given by Luzin. Luzin's example, which is essentially a subset of the Baire space, is based on the natural partial order on naturals, i.e. division. It consists of sequences which contain increasing subsequence in given order. We consider a variety of sets defined in a similar way. Some of them occurs to be Borel subsets of the Baire space, while others are analytic complete, hence not Borel. In particular, we show that an analogon of Luzin example based on the natural linear order on rationals is analytic complete. We also characterise all countable linear orders having such property.
In this paper we investigate the action of Polish groups (not necessary abelian) on an uncountable Polish spaces. We consider two main situations. First, when the orbits given by group action are small and the second when the family of orbits are at most countable. We have found some subgroups which are not measurable with respect to a given $σ$-ideals on the group and the action on some subsets gives a completely nonmeasurable sets with respect to some $σ$-ideals with a Borel base on the Polish space. In most cases the general results are consistent with ZFC theory and are strictly connected with cardinal coefficients. We give some suitable examples, namely the subgroup of isometries of the Cantor space where the orbits are sufficiently small. In the opposite case we give an example of the group of the homeomorphisms of a Polish space in which there is a large orbit and we have found the subgroup without Baire property and a subset of the mentioned space such that the action of this subgroup on this set is completely nonmeasurable set with respect to the $σ$-ideal of the subsets of first category.
Using a game-theoretic approach we present a generalization of the classical result of Brzuchowski, Cichoń, Grzegorek and Ryll-Nardzewski on non-measurable unions. We also present applications of obtained results to Marczewski--Burstin representable ideals, as well as to establishing some countability and continuity properties of measurable functions and homomorphisms between topological groups.
A $σ$-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 variants of this property and study their connections with countable chain condition, maximality and how well they are preserved via Fubini products.
Two-dimensional version of the classical Mycielski theorem says that for every comeager or conull set $X\subseteq [0,1]^2$ there exists a perfect set $P\subseteq [0,1]$ such that $P\times P\subseteq X\cup Δ$. We consider generalizations of this theorem by replacing a perfect square with a rectangle $A\times B$, where $A$ and $B$ are bodies of other types of trees with $A\subseteq B$. In particular, we show that for every comeager $G_δ$ set $G\subseteq ω^ω\times ω^ω$ there exist a Miller tree $M$ and a uniformly perfect tree $P\subseteq M$ such that $[P]\times [M]\subseteq G\cupΔ$ and that $P$ cannot be a Miller tree. In the case of measure we show that for every subset $F$ of $2^ω\times 2^ω$ of full measure there exists a uniformly perfect tree $P\subseteq 2^{<ω}$ such that $[P]\times[P]\subseteq F\cupΔ$ and no side of such a rectangle can be a body of a Silver tree or a Miller tree. We also show some properties of forcing extensions of the real line from which we derive nonstandard proofs of Mycielski-like theorems via Shoenfield Absoluteness Theorem.
In this paper we consider a notion of nonmeasurablity with respect to Marczewski and Marczewski-like tree ideals $s_0$, $m_0$, $l_0$, and $cl_0$. We show that there exists a subset $A$ of the Baire space $ω^ω$ which is $s$-, $l$-, and $m$-nonmeasurable, that forms dominating m.e.d. family. We introduce and investigate a notion of $\mathbb{T}$-Bernstein sets - sets that intersect but does not containt any body of a tree from a given family of trees $\mathbb{T}$. We also acquire some results on $\mathcal{I}$-Luzin sets, namely we prove that there are no $m_0$-, $l_0$-, and $cl_0$-Luzin sets and that if $\mathfrak{c}$ is a regular cardinal, then the algebraic sum (considered on the real line $\mathbb{R}$) of a generalized Luzin set and a generalized Sierpiński set belongs to $s_0, m_0$, $l_0$ and $cl_0$.
In this paper we consider a notion of $\mathcal{I}$-Luzin set which generalizes the classical notion of Luzin set and Sierpi{ń}ski set on Euclidean spaces. We show that there is a translation invariant $σ$-ideal $\mathcal{I}$ with Borel base for which $\mathcal{I}$-Luzin set can be $\mathcal{I}$-measurable. If we additionally assume that $\mathcal{I}$ has Smital property (or its weaker version) then $\mathcal{I}$-Luzin sets are $\mathcal{I}$-nonmeasurable. We give some constructions of $\mathcal{I}$-Luzin sets involving additive structure of $\mathbb{R}^n$. Moreover, we show that if $L$ is a Luzin set and $S$ is a Sierpi{ń}ski set then the complex sum $L+S$ cannot be a Bernstein set.
In this paper we shall introduce some nonmeasurable and completely nonmeasurable subsets of the plane with various additional properties, e.g. being Hamel basis, intersecting each line in a strong Luzin / Sierpiński set. Also some additive properties of Luzin and Sierpiński sets and their generalization I-Luzin sets, on the line are investigated.
We study and classify topologically invariant sigma-ideals with an analytic base on Euclidean spaces and evaluate the cardinal characteristics of such ideals.