arXiv subjects

Tristan van der Vlugt

Publications and source records attributed to Tristan van der Vlugt.

6 recordsLinked to original sources

arXiv2026

The Higher Closed Null Ideal(s)

We generalise the $σ$-ideal $\mathcal E$ generated by the closed Lebesgue null subsets of the Cantor space ${}^ω2$ to the context of higher Cantor spaces ${}^κ2$ for an uncountable cardinal $κ$ satisfying $κ=κ^{<κ}$. We introduce candidates for a higher closed null ideal based on slaloms on partitions of $κ$ into sets of size ${<}κ$, and study their combinatorial properties. We also give some relations between the cardinal characteristics of the higher closed null ideals and other cardinal characteristics.

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arXiv2026

Topology and category for singular product spaces

For $κ$ a regular uncountable cardinal, the higher Baire and Cantor spaces ${}^κκ$ and ${}^\kappa2$ (endowed with the ${<}κ$-box topology) have been relatively well-studied, but less is known about the case where $κ$ is singular. We will consider several spaces of functions and box topologies that could serve as higher Baire and Cantor spaces for singular cardinals. The ultimate focus of the article lies in studying cardinal characteristics of the ideal of $κ$-meagre subsets of these spaces.

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arXiv2025

Subseries Numbers for Convergent Subseries

Every conditionally convergent series of real numbers has a subseries that diverges. The subseries numbers, previously studied in arXiv:1801.06206 , answer the question how many subsets of the natural numbers are necessary, such that every conditionally convergent series has a subseries that diverges, with the index set being one of our chosen sets. By restricting our attention to subseries generated by an index set that is both infinite and coinfinite, we may ask the question where the subseries have to be convergent. The answer to this question is a cardinal characteristic of the continuum. We consider several closely related variations to this question, and show that our cardinal characteristics are related to several well-known cardinal characteristics of the continuum. In our investigation, we simultaneously will produce dual results, answering the question how many conditionally convergent series one needs such that no single infinite coinfinite set of indices makes all of the series converge.

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arXiv2025

The Horizontal Direction

We give a survey of cardinal charcteristics of the higher Cichoń diagram defined on the higher Baire space ${}^κκ$ for $κ$ regular with $2^{<κ}=κ$. Specifically, we will compare consistency proofs from the classical Cichoń diagram with various well-known forcing notions to similar constructions generalised to the higher Cichoń diagram. We are especially interested in separation in a horizontal direction, that is, the consistency of $\mathrm{add}(\mathcal M_κ)<\mathrm{non}(\mathcal M_κ)$ and of $\mathrm{cov}(\mathcal M_κ)<\mathrm{cof}(\mathcal M_κ)$. We will have a look at (higher analogues of) Cohen, Hechler, localisation, eventually different, Sacks, random, Laver, Mathias and Miller forcing, and their effect on the cardinal characteristics of the higher Cichoń diagram.

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arXiv2023

Cardinal Characteristics on Bounded Generalised Baire Spaces

We will give an overview of four families of cardinal characteristics defined on subspaces $\prod_{α\inκ}b(α)$ of the generalised Baire space ${}^κκ$, where $κ$ is strongly inaccessible and $b\in{}^κκ$. The considered families are bounded versions of the dominating, eventual difference, localisation and anti-localisation numbers, and their dual cardinals. We investigate parameters for which these cardinals are non-trivial and how the cardinals relate to each other and to other cardinals of the generalised Cichoń diagram. Finally we prove that different choices of parameters may lead to consistently distinct cardinals.

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arXiv2022

Separating Many Localisation Cardinals on the Generalised Baire Space

Given a cofinal cardinal function $h\in{}^κκ$ for $κ$ inaccessible, we consider the dominating $h$-localisation number, that is, the least cardinality of a dominating set of $h$-slaloms such that every $κ$-real is localised by a slalom in the dominating set. It was proved in arXiv:1611.08140 that the dominating localisation numbers can be consistently different for two functions $h$ (the identity function and the power function). We will construct a $κ$-sized family of functions $h$ and their corresponding localisation numbers, and use a ${\leq}κ$-supported product of a cofinality-preserving forcing to prove that any simultaneous assignment of these localisation numbers to cardinals above $κ$ is consistent. This answers an open question from arXiv:1611.08140 .

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