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Adam Kwela

Publications and source records attributed to Adam Kwela.

At least 37 records · Page 2Linked to original sources
arXiv2025

More on yet another ideal version of the bounding number

This is a continuation of the paper [J. Symb. Log. 87 (2022), 1065--1092]. For an ideal $\mathcal{I}$ on $ω$ we denote $\mathcal{D}_{\mathcal{I}}=\{f\inω^ω: f^{-1}[\{n\}]\in\mathcal{I} \text{ for every $n\in ω$}\}$ and write $f\leq_{\mathcal{I}} g$ if $\{n\inω:f(n)>g(n)\}\in\mathcal{I}$, where $f,g\inω^ω$. We study the cardinal numbers $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{\mathcal{I}} \times \mathcal{D}_{\mathcal{I}}))$ describing the smallest sizes of subsets of $\mathcal{D}_{\mathcal{I}}$ that are unbounded from below with respect to $\leq_{\mathcal{I}}$. In particular, we examine the relationships of $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{\mathcal{I}} \times \mathcal{D}_{\mathcal{I}}))$ with the dominating number $\mathfrak{d}$. We show that, consistently, $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{\mathcal{I}} \times \mathcal{D}_{\mathcal{I}}))>\mathfrak{d}$ for some ideal $\mathcal{I}$, however $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{\mathcal{I}} \times \mathcal{D}_{\mathcal{I}}))\leq\mathfrak{d}$ for all analytic ideals $\mathcal{I}$. Moreover, we give example of a Borel ideal with $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{\mathcal{I}} \times \mathcal{D}_{\mathcal{I}}))=add(\mathcal{M})$.

math.LO↗
arXiv2023

New Hindman spaces

We introduce a method that allows to turn topological questions about Hindman spaces into purely combinatorial questions about the Katětov order of ideals on $\mathbb{N}$. We also provide two applications of the method. (1) We characterize $F_σ$ ideals $\mathcal{I}$ for which there is a Hindman space which is not an $\mathcal{I}$-space under the continuum hypothesis. This reduces a topological question of Albin L. Jones about consistency of existence of a Hindman space which is not van der Waerden to the question whether the ideal of all non AP-sets is not below the ideal of all non IP-sets in the Katětov order. (2) Under the continuum hypothesis, we construct a Hindman space which is not an $\mathcal{I}_{1/n}$-space. This answers a question posed by Jana Flašková at the 22nd Summer Conference on Topology and its Applications.

math.GN↗
arXiv2023

Characterizing existence of certain ultrafilters

Following Baumgartner [J. Symb. Log. 60 (1995), no. 2], for an ideal $\mathcal{I}$ on $ω$, we say that an ultrafilter $\mathcal{U}$ on $ω$ is an $\mathcal{I}$-ultrafilter if for every function $f:ω\toω$ there is $A\in \mathcal{U}$ with $f[A]\in \mathcal{I}$. If there is an $\mathcal{I}$-ultrafilter which is not a $\mathcal{J}$-ultrafilter, then $\mathcal{I}$ is not below $\mathcal{J}$ in the Katětov order $\leq_{K}$ (i.e. for every function $f:ω\toω$ there is $A\in \mathcal{I}$ with $f^{-1}[A]\notin \mathcal{J}$). On the other hand, in general $\mathcal{I}\not\leq_{K}\mathcal{J}$ does not imply that existence of an $\mathcal{I}$-ultrafilter which is not a $\mathcal{J}$-ultrafilter is consistent. We provide some sufficient conditions on ideals to obtain the equivalence: $\mathcal{I}\not\leq_{K}\mathcal{J}$ if and only if it is consistent that there exists an $\mathcal{I}$-ultrafilter which is not a $\mathcal{J}$-ultrafilter. In some cases when the Katětov order is not enough for the above equivalence, we provide other conditions for which a similar equivalence holds. We are mainly interested in the cases when the family of all $\mathcal{I}$-ultrafilters or $\mathcal{J}$-ultrafilters coincides with some known family of ultrafilters: P-points, Q-points or selective ultrafilters (a.k.a. Ramsey ultrafilters). In particular, our results provide a characterization of Borel ideals $\mathcal{I}$ which can be used to characterize P-points as $\mathcal{I}$-ultrafilters. Moreover, we introduce a cardinal invariant which is used to obtain a sufficient condition for the existence of an $\mathcal{I}$-ultrafilter which is not a $\mathcal{I}$-ultrafilter. Finally, we prove some new results concerning existence of certain ultrafilters under various set-theoretic assumptions.

math.LO↗
arXiv2023

Spaces not distinguishing ideal pointwise and $σ$-uniform convergence

We examine topological spaces not distinguishing ideal pointwise and ideal $σ$-uniform convergence of sequences of real-valued continuous functions defined on them. For instance, we introduce a purely combinatorial cardinal characteristic (a sort of the bounding number $\mathfrak{b}$) and prove that it describes the minimal cardinality of topological spaces which distinguish ideal pointwise and ideal $σ$-uniform convergence. Moreover, we provide examples of topological spaces (focusing on subsets of reals) that do or do not distinguish the considered convergences. Since similar investigations for ideal quasi-normal convergence instead of ideal $σ$-uniform convergence have been performed in literature, we also study spaces not distinguishing ideal quasi-normal and ideal $σ$-uniform convergence of sequences of real-valued continuous functions defined on them.

math.GN↗
arXiv2023

Yet another ideal version of the bounding number

Let $\mathcal{I}$ be an ideal on $ω$. For $f,g\inω^ω$ we write $f \leq_{\mathcal{I}} g$ if $f(n) \leq g(n)$ for all $n\inω\setminus A$ with some $A\in\mathcal{I}$. Moreover, we denote $\mathcal{D}_{\mathcal{I}}=\{f\inω^ω: f^{-1}[\{n\}]\in\mathcal{I} \text{ for every $n\in ω$}\}$ (in particular, $\mathcal{D}_{Fin}$ denotes the family of all finite-to-one functions). We examine cardinal numbers $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{\mathcal{I}} \times \mathcal{D}_{\mathcal{I}}))$ and $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{Fin}\times \mathcal{D}_{Fin}))$ describing the smallest sizes of unbounded from below with respect to the order $\leq_{\mathcal{I}}$ sets in $\mathcal{D}_{Fin}$ and $\mathcal{D}_{\mathcal{I}}$, respectively. For a maximal ideal $\mathcal{I}$, these cardinals were investigated by M. Canjar in connection with coinitial and cofinal subsets of the ultrapowers. We show that $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{Fin} \times \mathcal{D}_{Fin})) =\mathfrak{b}$ for all ideals $\mathcal{I}$ with the Baire property and that $\aleph_1 \leq \mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{\mathcal{I}} \times \mathcal{D}_{\mathcal{I}})) \leq\mathfrak{b}$ for all coanalytic weak P-ideals (this class contains all $Π^0_4$ ideals). What is more, we give examples of Borel (even $Σ^0_2$) ideals $\mathcal{I}$ with $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{\mathcal{I}} \times \mathcal{D}_{\mathcal{I}}))=\mathfrak{b}$ as well as with $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{\mathcal{I}} \times \mathcal{D}_{\mathcal{I}})) =\aleph_1$.

math.LO↗
arXiv2023

Katětov order between Hindman, Ramsey, van der Waerden and summable ideals

A family I of subsets of a set X is an ideal on X if it is closed under taking subsets and finite unions of its elements. An ideal I on X is below an ideal J on Y in the Katetov order if there is a function $f:Y\to X$ such that $f^{-1}[A]\in J$ for every $A\in I$. We show that the Hindman ideal, the Ramsey ideal and the summable ideal are pairwise incomparable in the Katetov order, where * the Ramsey ideal consists of those sets of pairs of natural numbers which do not contain a set of all pairs of any infinite set (equivalently do not contain, in a sense, any infinite complete subgraph), * the Hindman ideal consists of those sets of natural numbers which do not contain any infinite set together with all finite sums of its members (equivalently do not contain IP-sets that are considered in Ergodic Ramsey theory), * the summable ideal consists of those sets of natural numbers such that the series of the reciprocals of its members is convergent. Moreover, we show that in the Katetov order the above mentioned ideals are not below the van der Waerden ideal that consists of those sets of natural numbers which do not contain arithmetic progressions of arbitrary finite length.

math.LO↗
arXiv2023

A unified approach to Hindman, Ramsey and van der Waerden spaces

For many years, there have been conducting research (e.g. by Bergelson, Furstenberg, Kojman, Kubiś, Shelah, Szeptycki, Weiss) into sequentially compact spaces that are, in a sense, topological counterparts of some combinatorial theorems, for instance Ramsey's theorem for coloring graphs, Hindman's finite sums theorem and van der Waerden's arithmetical progressions theorem. These spaces are defined with the aid of different kinds of convergences: IP-convergence, R-convergence and ordinary convergence. The first aim of this paper is to present a unified approach to these various types of convergences and spaces. Then, using this unified approach, we prove some general theorems about existence of the considered spaces and show that all results obtained so far in this subject can be derived from our theorems. The second aim of this paper is to obtain new results about the specific types of these spaces. For instance, we construct a Hausdorff Hindman space that is not an $\I_{1/n}$-space and a Hausdorff differentially compact space that is not Hindman. Moreover, we compare Ramsey spaces with other types of spaces. For instance, we construct a Ramsey space that is not Hindman and a Hindman space that is not Ramsey. The last aim of this paper is to provide a characterization that shows when there exists a space of one considered type that is not of the other kind. This characterization is expressed in purely combinatorial manner with the aid of the so-called Katětov order that has been extensively examined for many years so far. This paper may interest the general audience of mathematicians as the results we obtain are on the intersection of topology, combinatorics, set theory and number theory.

math.GN↗
arXiv2023

The ideal test for the divergence of a series

We generalize the classical Olivier's theorem which says that for any convergent series $\sum_n a_n$ with positive nonincreasing real terms the sequence $(n a_n)$ tends to zero. Our results encompass many known generalizations of Olivier's theorem and give some new instances. The generalizations are done in two directions: we either drop the monotonicity assumption completely or we relax it to the monotonicity on a large set of indices. In both cases, the convergence of $(na_n)$ is replaced by ideal convergence. In the second part of the paper, we examine families of sequences for which the assertions of our generalizations of Olivier's theorem fail. Here, we are interested in finding large linear and algebraic substructures in these families.

math.CA↗
arXiv2021

Density-Like and Generalized Density Ideals

We show that there exist uncountably many (tall and nontall) pairwise nonisomorphic density-like ideals on $ω$ which are not generalized density ideals. In addition, they are nonpathological. This answers a question posed by Borodulin-Nadzieja, Farkas, and Plebanek in [J. Symb. Log. \textbf{80} (2015), 1268--1289]. Lastly, we provide sufficient conditions for a density-like ideal to be necessarily a generalized density ideal.

math.FA↗
arXiv2020

Differentiability of continuous functions in terms of Haar-smallness

One of the classical results concerning differentiability of continuous functions states that the set $\mathcal{SD}$ of somewhere differentiable functions (i.e., functions which are differentiable at some point) is Haar-null in the space $C[0,1]$. By a recent result of Banakh et al., a set is Haar-null provided that there is a Borel hull $B\supseteq A$ and a continuous map $f\colon \{0,1\}^\mathbb{N}\to C[0,1]$ such that $f^{-1}[B+h]$ is Lebesgue's null for all $h\in C[0,1]$. We prove that $\mathcal{SD}$ is not Haar-countable (i.e., does not satisfy the above property with "Lebesgue's null" replaced by "countable", or, equivalently, for each copy $C$ of $\{0,1\}^\mathbb{N}$ there is an $h\in C[0,1]$ such that $\mathcal{SD}\cap (C+h)$ is uncountable. Moreover, we use the above notions in further studies of differentiability of continuous functions. Namely, we consider functions differentiable on a set of positive Lebesgue's measure and functions differentiable almost everywhere with respect to Lebesgue's measure. Furthermore, we study multidimensional case, i.e., differentiability of continuous functions defined on $[0,1]^k$. Finally, we pose an open question concerning Takagi's function.

math.FA↗
arXiv2019

Haar-smallest sets

In this paper we are interested in the following notions of smallness: a subset $A$ of an abelian Polish group $X$ is called Haar-countable/Haar-finite/Haar-$n$ if there are a Borel hull $B\supseteq A$ and a copy $C$ of $2^ω$ such that $(C+x)\cap B$ is countable/finite/of cardinality at most $n$, for all $x\in X$. Recently, Banakh et al. have unified the notions of Haar-null and Haar-meager sets by introducing Haar-$\mathcal{I}$ sets, where $\mathcal{I}$ is a collection of subsets of $2^ω$. It turns out that if $\mathcal{I}$ is the $σ$-ideal of countable sets, the ideal of finite sets or the collection of sets of cardinality at most $n$, then we get the above notions. Moreover, those notions have been studied independently by Zakrzewski (under a different name -- perfectly $κ$-small sets). We study basic properties of the corresponding families of small sets, give suitable examples distinguishing them (in all abelian Polish groups of the form $\mathbb{R}\times X$) and study $σ$-ideals generated by compact members of the considered families. In particular, we show that Haar-countable sets do not form an ideal. Moreover, we answer some questions concerning null-finite sets, asked by Banakh and Jabłońska, and pose several open problems.

math.FA↗
arXiv2019

Properties of simple density ideals

Let $G$ consist of all functions $g \colon ω\to [0,\infty)$ with $g(n) \to \infty$ and $\frac{n}{g(n)} \nrightarrow 0$. Then for each $g\in G$ the family $\mathcal{Z}_g=\{A\subseteqω:\ \lim_{n\to\infty}\frac{\text{card}(A\cap n)}{g(n)}=0\}$ is an ideal associated to the notion of so-called upper density of weight $g$. Although those ideals have recently been extensively studied, they do not have their own name. In this paper, for Reader's convenience, we propose to call them simple density ideals. We show that there are $\mathfrak{c}$ many non-isomorphic (in fact even incomparable with respect to Katětov order) simple density ideals. Moreover, we prove that for a given $A\subset G$ with $\text{card}(A)<\mathfrak{b}$ one can construct a family of cardinality $\mathfrak{c}$ of pairwise incomparable (with respect to inclusion) simple density ideals which additionally are incomparable with all $\mathcal{Z}_g$ for $g\in A$. We show that this cannot be generalized to Katětov order as the ideal $\mathcal{Z}$ of sets of asymptotic density zero is maximal in the sense of Katětov order among all simple density ideals. We examine how many substantially different functions $g$ can generate the same ideal $\mathcal{Z}_g$ -- it turns out that the answer is either $1$ or $\mathfrak{c}$ (depending on $g$).

math.FA↗
arXiv2018

Ideal weak QN-spaces

This paper is devoted to studies of IwQN-spaces and some of their cardinal characteristics. Recently, Šupina proved that I is not a weak P-ideal if and only if any topological space is an IQN-space. Moreover, under $\mathfrak{p}=\mathfrak{c}$ he constructed a maximal ideal I (which is not a weak P-ideal) for which the notions of IQN-space and QN-space do not coincide. In this paper we show that, consistently, there is an ideal I (which is not a weak P-ideal) for which the notions of IwQN-space and wQN-space do not coincide. We also prove that for this ideal the ideal version of Scheepers Conjecture does not hold (this is the first known example of such weak P-ideal). We obtain a strictly combinatorial characterization of ${\tt non}(\text{IwQN-space})$ similar to the one given by Šupina in the case of ${\tt non}(\text{IQN-space})$. We calculate ${\tt non}(\text{IQN-space})$ and ${\tt non}(\text{IwQN-space})$ for some weak P-ideals. Namely, we show that $\mathfrak{b}\leq{\tt non}(\text{IQN-space})\leq{\tt non}(\text{IwQN-space})\leq\mathfrak{d}$ for every weak P-ideal I and that ${\tt non}(\text{IQN-space})={\tt non}(\text{IwQN-space})=\mathfrak{b}$ for every $\mathtt{F_σ}$ ideal I as well as for every analytic P-ideal I generated by an unbounded submeasure (this establishes some new bounds for $\mathfrak{b}(I,I,Fin)$). As a consequence, we obtain some bounds for ${\tt add}(\text{IQN-space})$. In particular, we get ${\tt add}(\text{IQN-space})=\mathfrak{b}$ for analytic P-ideals I generated by an unbounded submeasure. By a result of Bukovský, Das and Šupina it is known that in the case of tall ideals I the notions of IQN-space (IwQN-space) and QN-space (wQN-space) cannot be distinguished. We prove that if I is a tall ideal and X is a topological space of cardinality less than ${\tt cov^*}(I)$, then X is an IwQN-space if and only if it is a wQN-space.

math.GN↗
arXiv2018

Erdős-Ulam ideals vs. simple density ideals

The main aim of this paper is to bridge two directions of research generalizing asymptotic density zero sets. This enables to transfer results concerning one direction to the other one. Consider a function $g\colonω\to [0,\infty)$ such that $\lim_{n\to\infty}g(n)=\infty$ and $\frac{n}{g(n)}$ does not converge to $0$. Then the family $\mathcal{Z}_g=\{A\subseteqω:\ \lim_{n\to\infty}\frac{\text{card}(A\cap n)}{g(n)}=0\}$ is an ideal called simple density ideal (or ideal associated to upper density of weight $g$). We compare this class of ideals with Erdős-Ulam ideals. In particular, we show that there are $\sqsubseteq$-antichains of size $\mathfrak{c}$ among Erdős-Ulam ideals which are and are not simple density ideals. We characterize simple density ideals which are Erdős-Ulam as those containing the classical ideal of sets of asymptotic density zero. We also characterize Erdős-Ulam ideals which are simple density ideals. In the latter case we need to introduce two new notions. One of them, called increasing-invariance of an ideal $\mathcal{I}$, asserts that given $B\in\mathcal{I}$ and $C\subseteqω$ with $\text{card}(C\cap n)\leq\text{card}(B\cap n)$ for all $n$, we have $C\in\mathcal{I}$. Finally, we pose some open problems.

math.CO↗
arXiv2016

Homogeneous ideals on countable sets

We say that an ideal I is homogeneous, if its restriction to any I-positive subset is isomorphic to I. The paper investigates basic properties of this notion -- we give examples of homogeneous ideals and present some applications to topology and ideal convergence. Moreover, we answer questions related to our research.

math.LO↗
arXiv2016

Ideal equal Baire classes

For any Borel ideal we characterize ideal equal Baire system generated by the families of continuous and quasi-continuous functions, i.e., the families of ideal equal limits of sequences of continuous and quasi-continuous functions.

math.GN↗
arXiv2015

Additivity of the ideal of microscopic sets

A set $M\subset\mathbb{R}$ is microscopic if for each $\varepsilon>0$ there is a sequence of intervals $(J_n)_{n\inω}$ covering $M$ and such that $|J_n|\leq \varepsilon^{n+1}$ for each $n\inω$. We show that there is a microscopic set which cannot be covered by a sequence $(J_n)_{n\inω}$ with $\{n\inω:J_n\neq\emptyset\}$ of lower asymptotic density zero. We prove (in ZFC) that additivity of the ideal of microscopic sets is $ω_1$. This solves a problem of G. Horbaczewska. Finally, we discuss additivity of some generalizations of this ideal.

math.LO↗