arXiv subjects

Diego A. Mejía

Publications and source records attributed to Diego A. Mejía.

At least 19 recordsLinked to original sources

arXiv2026

Directed schemes of ideals and cardinal characteristics, I: the meager additive ideal

We introduce the notion of directed scheme of ideals to characterize peculiar ideals on the reals, which comes from a formalization of the framework of Yorioka ideals for strong measure zero sets. We prove general theorems for directed schemes and propose a directed scheme $\vec{\mathcal{M}} = \{\mathcal{M}_I \colon I\in\mathbb{I}\}$ for the ideal $\mathcal{MA}$ of meager-additive sets of reals. This directed scheme does not only helps us to understand more the combinatorics of $\mathcal{MA}$ and its cardinal characteristics, but provides us new characterizations of the additivity and cofinality numbers of the meager ideal of the reals. In addition, we display connections between the characteristics associated with $\mathcal{M}_I$ and other classical characteristics. Furthermore, we demonstrate the consistency of $\mathrm{cov}(\mathcal{NA})<\mathfrak{c}$ and $\mathrm{cof}(\mathcal{MA})<\mathrm{non}(\mathcal{SN})$. The first one answers a question raised by the authors in arXiv:2401.15364.

math.LO↗

arXiv2025

Finitely additive measures on Boolean algebras

In this article, we conduct a detailed study of \emph{finitely additive measures} (fams) in the context of Boolean algebras, focusing on three specific topics: freeness and approximation, existence and extension criteria, and integration theory. In the first topic, we present a classification of \emph{free} finitely additive measures, that is, those for which the measure of finite sets is zero, in terms of approximation to uniform probability measures. This inspires a weaker version of this notion, which we call the \emph{uniform approximation property}, characterized in terms of freeness and another well-determined type of fams we call \emph{uniformly supported}. In the second topic, we study new criteria for the existence and extension of finitely additive measures. In particular, we provide an extension of the so-called \emph{compatibility theorem} -- which characterizes when two finitely additive measures can be extended -- yielding a precise and compact characterization of when three finitely additive measures can be simultaneously extended, under the assumption that one of them is an ultrafilter. Finally, we study a Riemann-type integration theory on fields of sets with respect to finitely additive measures, allowing us to extend and generalize some classical concepts and results from real analysis, such as Riemann integration over rectangles in $\mathbb{R}^{n}$ and the Jordan measure. We also generalize the extension criteria for fams allowing desired values of integrals of a given set of functions. At the end, we explore the connection between integration in fields of sets and the Lebesgue integration in the Stone space of the corresponding field, where we establish a characterization of integrability in the sense of the Lebesgue-Vitali theorem, which follows as a consequence of our results.

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arXiv2025

Uniformity numbers of the null-additive and meager-additive ideals

Denote by $\mathcal{NA}$ and $\mathcal{MA}$ the ideals of null-additive and meager-additive subsets of~$2^ω$, respectively. We prove in ZFC that $\mathrm{add}(\mathcal{NA})=\mathrm{non}(\mathcal{NA})$ and introduce a new (Polish) relational system to reformulate Bartoszyński's and Judah's characterization of the uniformity of $\mathcal{MA}$, which is helpful to understand the combinatorics of $\mathcal{MA}$ and to prove consistency results. As for the latter, we prove that $\mathrm{cov}(\mathcal{MA})<\mathfrak{c}$ (even $\mathrm{cov}(\mathcal{MA})<\mathrm{non}(\mathcal{N})$) is consistent with ZFC, as well as several constellations of Cichoń's diagram with $\mathrm{non}(\mathcal{NA})$, $\mathrm{non}(\mathcal{MA})$ and $\mathrm{add}(\mathcal{SN})$, which include $\mathrm{non}(\mathcal{NA})<\mathfrak{b}< \mathrm{non}(\mathcal{MA})$ and $\mathfrak{b}< \mathrm{add}(\mathcal{SN})<\mathrm{cov}(\mathcal{M})<\mathfrak{d}=\mathfrak{c}$.

math.LO↗

arXiv2025

Separating cardinal characteristics of the strong measure zero ideal

Let $\mathcal{SN}$ be the $σ$-ideal of the strong measure zero sets of reals. We present general properties of forcing notions that allow to control of the additivity of $\mathcal{SN}$ after finite support iterations. This is applied to force that the four cardinal characteristics associated with $\mathcal{SN}$ are pairwise different: \[\mathrm{add}(\mathcal{SN})<\mathrm{cov}(\mathcal{SN})<\mathrm{non}(\mathcal{SN})<\mathrm{cof}(\mathcal{SN}).\] Furthermore, we construct a forcing extension satisfying the above and Cichoń's maximum (i.e.\ that the non-dependent values in Cichoń's diagram are pairwise different).

math.LO↗

arXiv2025

Forcing techniques for Cichoń's Maximum: Lecture notes for the mini-course at the University of Vienna

Cichoń's diagram describes the connections between combinatorial notions related to measure, category, and compactness of sets of irrational numbers. In the second part of the 2010's, Goldstern, Kellner and Shelah constructed a forcing model of Cichoń's Maximum (meaning that all non-dependent cardinal characteristics are pairwise different) by using large cardinals. Some years later, we eliminated this large cardinal assumption. In this mini-course, we explore the forcing techniques to construct the Cichoń's Maximum model and much more.

math.LO↗

arXiv2025

The measure algebra adding $θ$-many random reals is $θ$-FAM-linked

The notion of $θ$-FAM-linkedness, introduced in the second author's master thesis, is a formalization of the notion of strong FAM limits for intervals, whose initial form and applications have appeared in the work of Saharon Shelah, Jakob Kellner, and Anda Tănasie, for controlling cardinals characteristics of the continuum in ccc forcing extensions. This generalization was successful in such a thesis to establish a general theory of iterated forcing using finitely additive measures. In this paper, using probability theory tools developed in the same thesis, we refine Saharon Shelah's proof of the fact that random forcing is $σ$-FAM-linked and prove that any complete Boolean algebra with a strictly positive probability measure satisfying the $θ$-density property is $θ$-FAM-linked. As a consequence, we get a new collection of examples of $θ$-FAM-linked forcing notions: the measure algebra adding $θ$-many random reals.

math.LO↗

arXiv2025

Probability Trees

In this article, we introduce a formal definition of the concept of probability tree and conduct a detailed and comprehensive study of its fundamental structural properties. In particular, we define what we term an inductive probability measure and prove that such trees can be identified with these measures. Furthermore, we prove that probability trees are completely determined by probability measures on the Borel $σ$-algebra of the tree's body. We then explore applications of probability trees in several areas of mathematics, including probability theory, measure theory, and set theory. In the first, we show that the cumulative distribution of finitely many dependent and non-identically distributed Bernouli tests is bounded by the cumulative distribution of some binomial distribution. In the second, we establish a close relationship between probability trees and the real line, showing that Borel, measurable sets, and their measures can be preserved, as well as other combinatorial properties. Finally, in set theory, we establish that the null ideal associated with suitable probability trees is Tukey equivalent to the null ideal on $[0, 1]$. This leads to a new elementary proof of the fact that the null ideal of a free $σ$-finite Borel measure on a Polish space is Tukey equivalent with the null ideal of $\mathbb{R}$, which supports that the associated cardinal characteristics remain invariant across the spaces in which they are defined.

math.PR↗

arXiv2024

More about the cofinality and the covering of the ideal of strong measure zero sets

We improve the previous work of Yorioka and the first author about the combinatorics of the ideal $\mathcal{SN}$ of strong measure zero sets of reals. We refine the notions of dominating systems of the first author and introduce the new combinatorial principle $\mathrm{DS}(δ)$ that helps to find simple conditions to deduce $\mathfrak{d}_κ\leq \mathrm{cof}(\mathcal{SN})$ (where $\mathfrak{d}_κ$ is the dominating number on $κ^κ$). In addition, we find a new upper bound of $\mathrm{cof}(\mathcal{SN})$ by using products of relational systems and cardinal characteristics associated with Yorioka ideals. In addition, we dissect and generalize results from Pawlikowski to force upper bounds of the covering of $\mathcal{SN}$, particularly for finite support iterations of precaliber posets. Finally, as applications of our main theorems, we prove consistency results about the cardinal characteristics associated with $\mathcal{SN}$ and the principle $\mathrm{DS}(δ)$. For example, we show that $\mathrm{cov}(\mathcal{SN})<\mathrm{non}(\mathcal{SN})=\mathfrak{c}<\mathrm{cof}(\mathcal{SN})$ holds in Cohen model, and we refine a result (and the proof) of the first author about the consistency of $\mathrm{cov}(\mathcal{SN})<\mathrm{non}(\mathcal{SN})<\mathrm{cof}(\mathcal{SN})$, with $\mathfrak{c}$ in any desired position with respect to $\mathrm{cof}(\mathcal{SN})$, and the improvement that $\mathrm{non}(\mathcal{SN})$ can be singular here.

math.LO↗

arXiv2024

A general theory of iterated forcing using finitely additive measures

Based on the work of Shelah, Kellner, and Tănasie (Fund. Math., 166(1-2):109-136, 2000 and Comment. Math. Univ. Carolin., 60(1):61-95, 2019), and the recent developments in the third author's master's thesis, we develop a general theory of iterated forcing using finitely additive measures. For this purpose, we introduce two new notions: on the one hand, we define a new linkedness property, called $μ$-$\mathrm{FAM}$-linked and, on the other hand, we generalize the notion of intersection number to forcing notions, which justifies the limit steps of our iteration theory. Our theory also generalizes iterations with ultrafilters, which have played an important role in the proof of the consistency of Cichoń's maximum. We further show that any iteration constructed with our theory preserves strong unbounded families and what we call anti-Bendixson families, which play a central role in preserving witnesses of $\mathrm{cov}(\mathcal{N})$ of singular size (even of countable cofinality). We also show that our iteration method does not increase $\mathrm{non}(\mathcal{E})$, the smallest size of a set of reals that cannot be covered by an $F_σ$ measure zero set. Finally, we apply our theory to prove a new separation of the left-hand side of Cichoń's diagram where $\mathrm{cov}(\mathcal{N})$ is possibly singular, even with countable cofinality.

math.LO↗

arXiv2024

Preservation of splitting families and cardinal characteristics of the continuum

We show how to construct, via forcing, splitting families than are preserved by a certain type of finite support iterations. As an application, we construct a model where 15 classical characteristics of the continuum are pairwise different, concretely: the 10 (non-dependent) entries in Cichoń's diagram, $\mathfrak{m}(2\text{-Knaster})$, $\mathfrak{p}$, $\mathfrak{h}$, the splitting number $\mathfrak{s}$ and the reaping number $\mathfrak{r}$.

math.LO↗

arXiv2024

Anatomy of $\tilde{\mathbb{E}}$

We present a detailed general framework to describe the forcing $\tilde{\mathbb{E}}$, defined by Kellner, Shelah and Tanăsie to prove the consistency with ZFC of an alternative order of Cichoń's maximum. Our presentation is close to the framework of tree-creature forcing notions from Horowitz and Shelah. We show that the posets in this class have strong FAM limits for intervals (in recent terminology, they are $σ$-FAM-linked) and, furthermore, that they also have strong ultrafilter limits for intervals.

math.LO↗

arXiv2022

Some notes about power residues modulo prime

Let $q$ be a prime. We classify the odd primes $p\neq q$ such that the equation $x^2\equiv q\pmod{p}$ has a solution, concretely, we find a subgroup $\mathbb{L}_{4q}$ of the multiplicative group $\mathbb{U}_{4q}$ of integers relatively prime with $4q$ (modulo $4q$) such that $x^2\equiv q\pmod{p}$ has a solution iff $p\equiv c\pmod{4q}$ for some $c\in\mathbb{L}_{4q}$. Moreover, $\mathbb{L}_{4q}$ is the only subgroup of $\mathbb{U}_{4q}$ of half order containing $-1$. Considering the ring $\mathbb{Z}[\sqrt{2}]$, for any odd prime $p$ it is known that the equation $x^2\equiv 2\pmod{p}$ has a solution iff the equation $x^2-2y^2=p$ has a solution in the integers. We ask whether this can be extended in the context of $\mathbb{Z}[\sqrt[n]{2}]$ with $n\geq 2$, namely: for any prime $p\equiv 1\pmod{n}$, is it true that $x^n\equiv 2\pmod{p}$ has a solution iff the equation $D^2_n(x_0,\ldots,x_{n-1})=p$ has a solution in the integers? Here $D^2_n(\bar{x})$ represents the norm of the field extension $\mathbb{Q}(\sqrt[n]{2})$ of $\mathbb{Q}$. We solve some weak versions of this problem, where equality with $p$ is replaced by $0\pmod{p}$ (divisible by $p$), and the "norm" $D^r_n(\bar{x})$ is considered for any $r\in\mathbb{Z}$ in the place of $2$.

math.NT↗

arXiv2022

Forcing constellations of Cichoń's diagram by using the Tukey order

We use known finite support iteration techniques to present various examples of models where several cardinal characteristics of Cichoń's diagram are pairwise different. We show some simple examples forcing the left-hand side of Cichoń's diagram, and present the technique of restriction to models to force Cichoń's maximum (original from Goldstern, Kellner, Shelah, and the second author). We focus on how the values forced in all the constellations are obtained via the Tukey order.

math.LO↗

arXiv2021

Controlling cardinal characteristics without adding reals

We investigate the behavior of cardinal characteristics of the reals under extensions that do not add new ${<}κ$-sequences (for some regular $κ$). As an application, we show that consistently the following cardinal characteristics can be different: The ("independent") characteristics in Cichoń's diagram, plus $\aleph_1<\mathfrak m<\mathfrak p<\mathfrak h<\mathrm{add}(\mathcal{N})$. (So we get thirteen different values, including $\aleph_1$ and continuum). We also give constructions to alternatively separate other MA-numbers (instead of $\mathfrak m$), namely: MA for $k$-Knaster from MA for $k+1$-Knaster; and MA for the union of all $k$-Knaster forcings from MA for precaliber.

math.LO↗

arXiv2020

Filter-linkedness and its effect on preservation of cardinal characteristics

We introduce the property ``$F$-linked'' of subsets of posets for a given free filter $F$ on the natural numbers, and define the properties ``$μ$-$F$-linked'' and ``$θ$-$F$-Knaster'' for posets in a natural way. We show that $θ$-$F$-Knaster posets preserve strong types of unbounded families and of maximal almost disjoint families. Concerning iterations of such posets, we develop a general technique to construct $θ$-$\mathrm{Fr}$-Knaster posets (where $\mathrm{Fr}$ is the Frechet ideal) via matrix iterations of ${<}θ$-ultrafilter-linked posets (restricted to some level of the matrix). This is applied to prove consistency results about Cichoń's diagram (without using large cardinals) and to prove the consistency of the fact that, for each Yorioka ideal, the four cardinal invariants associated with it are pairwise different. At the end, we show that three strongly compact cardinals are enough to force that Cichoń's diagram can be separated into $10$ different values.

math.LO↗

arXiv2020

Controlling classical cardinal characteristics while collapsing cardinals

Given a forcing notion $P$ that forces certain values to several classical cardinal characteristics of the reals, we show how we can compose $P$ with a collapse (of a cardinal $λ>κ$ to $κ$) such that the composition still forces the previous values to these characteristics. We also show how to force distinct values to $\mathfrak m$, $\mathfrak p$ and $\mathfrak h$ and also keeping all the values in Cichoń's diagram distint, using the Boolean Ultrapower method of arXiv:1708.03691 . (In arXiv:2006.09826 , the same was done for the newer Cichoń's Maximum construction, which avoids large cardinals.)

math.LO↗

arXiv2020

Cichoń's maximum without large cardinals

Cichoń's diagram lists twelve cardinal characteristics (and the provable inequalities between them) associated with the ideals of null sets, meager sets, countable sets, and $σ$-compact subsets of the irrationals. It is consistent that all entries of Cichoń's diagram are pairwise different (apart from $\textrm{add}(\mathcal{M})$ and $\textrm{cof}(\mathcal{M})$, which are provably equal to other entries). However, the consistency proofs so far required large cardinal assumptions. In this work, we show the consistency without such assumptions.

math.LO↗

arXiv2019

The covering number of the strong measure zero ideal can be above almost everything else

We show that certain type of tree forcings, including Sacks forcing, increases the covering of the strong measure zero ideal $\mathcal{SN}$. As a consequence, in Sacks model, such covering number is equal to the size of the continuum, which indicates that this covering number is consistently larger than any other classical cardinal invariant of the continuum. Even more, Sacks forcing can be used to force that $\mathrm{non}(\mathcal{SN})<\mathrm{cov}(\mathcal{SN})<\mathrm{cof}(\mathcal{SN})$, which is the first consistency result where more than two cardinal invariants associated with $\mathcal{SN}$ are pairwise different. Another consequence is that $\mathcal{SN}\subseteq s^0$ in ZFC where $s^0$ denotes the Marczewski's ideal.

math.LO↗