Tomek Bartoszynski, Masaru Kada
We prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there is a basis of the meager ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler's classical result in the theory of forcing.
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We show that the set of the ground-model reals has strong measure zero (is strongly meager) after adding a single Cohen real (random real). As consequence we prove that the set of the ground-model reals has strong measure zero after adding a single Hechler real.
We prove the following theorem: For a partially ordered set Q such that every countable subset has a strict upper bound, there is a forcing notion satisfying ccc such that, in the forcing model, there is a basis of the null ideal of the real line which is order-isomorphic to Q with respect to set-inclusion. This is a variation of Hechler's classical result in the theory of forcing, and the statement of the theorem for the meager ideal has been already proved by Bartoszynski and the author.
A set X subseteq R is strongly meager if for every measure zero set H, X+H not= R. Let SM denote the collection of strongly meager sets. We show that assuming CH, SM is not an ideal.
We study the ideal of meager sets and related ideals.
We show that the following are consistent with ZFC: 1. Strongly meager sets form an ideal with the same additivity as the ideal of meager sets. 2. There exists a strong measure zero set of size > d (dominating number).
We show that if the real line is the disjoint union of kappa meager sets such that every meager set is contained in a countable union of them, then kappa = omega_1. This answers a question addressed by J. Cichon. We also prove two theorems saying roughly that any attempt to produce the isomorphism type of the meager ideal in the Cohen real and the random real extensions must fail. All our results hold for "meager" replaced by "null", as well.
Directed schemes of ideals and cardinal characteristics, I: the meager additive ideal
本文引入方向理想方案以刻画实数上的特殊理想，提出方向方案$\vec{\mathcal{M}}$用于描述meager加法理想，探讨其组合学及基数特征，并证明cov(NA)<c与cof(MA)<non(SN)的一致性。
Combinatorics of translations of meager and closed measure zero sets
研究贫集与闭测度零集平移的组合性质，探讨相关Borel猜想，获得贫理想可加性新刻画，并证明若干基数不变量关系。
Cardinal invariants associated with the combinatorics of the uniformity number of the ideal of meager-additive sets
本文综述了与贫瘠可加集理想均匀数相关的基数不变量 $\mathfrak{b}_b^\mathsf{eq}$ 及其对偶 $\mathfrak{d}_b^\mathsf{eq}$，探讨它们与 Cichoń 图中基数的联系，并提出若干公开问题。
We study the relationship between the sigma-ideal generated by closed measure zero sets and the ideals of null and meager sets. We show that the additivity of the ideal of closed measure zero sets is not bigger than covering for category. As a consequence we get that the additivity of the ideal of closed measure zero sets is equal to the additivity of the ideal of meager sets.
Uniformity numbers of the null-additive and meager-additive ideals
本文研究零可加与贫可加理想的基数特征，证明二者加性与一致数相等，并借助新的 Polish 关系系统建立多种 Cichoń 图谱构型的相容性。
On Star operation and some ideals on the Baire space
本文研究了Baire空间上的几个σ-理想，引入并分析了理想G和SMZ⁺，建立了它们与经典理想的结构关系，计算了M₋的基数不变量，并证明了星运算下的对偶关系，最后证明了M₋的树二分定理。
The main result of this paper is a partial answer to [math.LO/9909115, Problem 5.5]: a finite iteration of Universal Meager forcing notions adds generic filters for many forcing notions determined by universality parameters. We also give some results concerning cardinal characteristics of the sigma-ideals determined by those universality parameters.
We show that some set-theoretic assumptions (for example Martin's Axiom) imply that there is no translation invariant Borel hull operation on the family of Lebesgue null sets and on the family of meager sets in (in R^n). We also prove that if the meager ideal admits a monotone Borel hull operation, then there is also a monotone Borel hull operation on the sigma-algebra of sets with the property of Baire.
A classical theorem due to Mycielski states that an equivalence relation $E$ having the Baire property and meager equivalence classes must have a perfect set of pairwise inequivalent elements. We consider equivalence relations with $I$-small equivalence classes, where $I$ is a proper $σ$-ideal, and ask whether they have a perfect set of pairwise inequivalent elements. We give a positive answer for $E$ universally Baire. We show that the answer for $E$ $\mathbf{Δ_{2}^{1}}$ is independent of $ZFC$, and find set theoretic assumptions equivalent to it when $I$ is the countable ideal. For equivalen…
The Category Dichotomy for Ideals
本文在ZFC框架下反驳了理想类的二元性，通过构造反例并探讨高P-理想与Q^{+}理想在不同模型中的性质。
A subset $X$ of a Polish group $G$ is \emph{Haar null} if there exists a Borel probability measure $μ$ and a Borel set $B$ containing $X$ such that $μ(gBh)=0$ for every $g,h \in G$. A set $X$ is \emph{Haar meager} if there exists a compact metric space $K$, a continuous function $f : K \to G$ and a Borel set $B$ containing $X$ such that $f^{-1}(gBh)$ is meager in $K$ for every $g,h \in G$. We calculate (in $ZFC$) the four cardinal invariants ($\rm add$, $\rm cov$, $\rm non$, $\rm cof$) of these two $σ$-ideals for the simplest non-locally compact Polish group, namely in the case $G = \mathbb{Z}…
Cardinal invariants of idealized Miller null sets
本文研究了理想化Miller空集的基数不变量，通过计算典型Borel理想参数化的$σ$-理想，扩展了Cichoń的最大值。
I prove forcing preservation theorems for products of definable partial orders preserving the cofinality of the meager or null ideal. Rectangular Ramsey theorems for related ideals follow from the proofs.
Lebesgue measure zero modulo ideals on the natural numbers
本文提出了实数Lebesgue测度零集理想模自然数上理想的重新表述，证明了其σ-理想性质及与Baire性质的等价关系，并研究了相关基数特征在Cichoń图中的位置及协调性结果。
We ask whether $\mathbf{Δ^1_2}$ or $\mathbf{Σ^1_2}$ equivalence relations with $I$-small classes for $I$ a $σ$-ideal must have perfectly many classes. We show that for a wide class of ccc $σ$-ideals, a positive answer for $\mathbf{Δ^1_2}$ equivalence relations is equivalent to the $I$-measurability of $\mathbf{Δ^1_2}$ sets. However, the analogous statement for $\mathbf{Σ^1_2}$ equivalence relations is false: $\mathbf{Σ^1_2}$ equivalence relations with meager classes have a perfect set of pairwise inequivalent elements if and only if $\mathbf{Δ^1_2}$ sets have the Baire property.
We define the property of Pi_2-compactness of a statement phi of set theory, meaning roughly that the hard core of the impact of phi on combinatorics of aleph_1 can be isolated in a canonical model for the statement phi. We show that the following statements are Pi_2-compact: ``dominating number = aleph_1,'' ``cofinality of the meager ideal = aleph_1'', ``cofinality of the null ideal = aleph_1'', existence of various types of Souslin trees and variations on uniformity of measure and category = aleph_1. Several important new metamathematical patterns among classical statements of set theory are…
Mad families of vector subspaces and the smallest nonmeager set of reals
本文证明参数化菱形原理（对应贫理想一致性）蕴含可数向量空间中无限极大几乎不相交块子空间族的最小基数为阿列夫一，进而在Miller模型中该不变量为阿列夫一。
A Set-Theoretic Translation of Modal Logic via Forcing
本文通过力迫法为模态逻辑的扩张理论建立集合论翻译，证明其可靠性与完备性，并构建相应的Kripke模型。
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