Saharon Shelah
We show that every null-additive set is meager-additive, where: (1) a set X subseteq 2^omega is null-additive if for every Lebesgue null set A subseteq 2^omega, X+A is null too; (2) we say that X subseteq 2^omega is meager-additive if for every A subseteq 2^omega which is meager also X+A is meager.
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We will show that there is no ZFC example of a set distinguishing between universally null and perfectly meager sets.
We develop a theory of \emph{sharp measure zero} sets that parallels Borel's \emph{strong measure zero}, and prove a theorem analogous to Galvin-Myscielski-Solovay Theorem, namely that a set of reals has sharp measure zero if and only if it is meager-additive. Some consequences: A subset of $2^ω$ is meager-additive if and only if it is $\mathcal E$-additive; if $f:2^ω\to2^ω$ is continuous and $X$ is meager-additive, then so is $f(X)$.
It is consistent that \[ \aleph_1 < \mathrm{add}(\mathrm{Null}) < \mathrm{add}(\mathrm{Meager})= \mathfrak{b} < \mathrm{cov}(\mathrm{Null}) < \mathrm{non}(\mathrm{Meager}) < \mathrm{cov}(\mathrm{Meager}) = 2^{\aleph_0}. \] Assuming four strongly compact cardinals, it is consistent that \[ \aleph_1 < \mathrm{add}(\mathrm{Null}) <\mathrm{add}(\mathrm{Meager})=\mathfrak{b} < \mathrm{cov}(\mathrm{Null}) < \mathrm{non}(\mathrm{Meager}) < \mathrm{cov}(\mathrm{Meager}) < \mathrm{non}(\mathrm{Null}) < \mathrm{cof}(\mathrm{Meager})= \mathfrak{d} < \mathrm{cof}(\mathrm{Null}) < 2^{\aleph_0}. \]
We show that it is consistent with ZFC that all filters which have the Baire property are Lebesgue measurable. We also show that the existence of a Sierpinski set implies that there exists a nonmeasurable filter which has the Baire property.
The paper contains two results pointing to the lack of symmetry between measure and category. Assume CH. There exists a strongly meager subset of the Cantor set that can be mapped onto the Cantor set by a uniformly continuous function. (It is well known that uniformly continuous image of a strongly null set is strongly null). A ZFC version of this result is also given.
We prove that it is relatively consistent with $\mathrm{ZFC}$ that every strong measure zero subset of the real line is meager-additive while there are uncountable strong measure zero sets (i.e., Borel's conjecture fails). This answers a long-standing question due to Bartoszyński and Judah.
A separable metric space X is an H-null set if any uniformly continuous image of X has Hausdorff dimension zero. upper H-null, directed P-null and P-null sets are defined likewise, with other fractal dimensions in place of Hausdorff dimension. We investigate these sets and show that in 2^ω they coincide, respectively, with strongly null, meager-additive, T' and null-additive sets. Some consequences: A subset of 2^ω is meager-additive if and only if it is E-additive; if f:2^ω->2^ω is continuous and X is meager-additive, then so is f(X), and likewise for null-additive and T'-sets.
We survey results about Haar null subsets of (not necessarily locally compact) Polish groups. The aim of this paper is to collect the fundamental properties of the various possible definitions of Haar null sets, and also to review the techniques that may enable the reader to prove results in this area. We also present several recently introduced ideas, including the notion of Haar meager sets, which are closely analogous to Haar null sets. We prove some results in a more general setting than that of the papers where they were originally proved and prove some results for Haar meager sets which …
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。
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。
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。
Countably perfectly meager and countably perfectly null sets
本文通过引入可数完全贫乏集和可数完全零集以强化普遍贫乏集与普遍零集概念，并在连续统假设弱化条件下证明了二者不等价。
We use a (countable support) creature construction to show that consistently \[ \mathfrak d=\aleph_1= \text{cov}(\text{NULL}) < \text{non}(\text{MEAGER}) < \text{non}(\text{NULL}) < \text{cof}(\text{NULL}) < 2^{\aleph_0}. \] The same method shows the consistency of \[ \mathfrak d=\aleph_1= \text{cov}(\text{NULL}) < \text{non}(\text{NULL}) < \text{non}(\text{MEAGER}) < \text{cof}(\text{NULL}) < 2^{\aleph_0}. \]
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}…
We are interested in subgroups of the reals that are small in one and large in another sense. We prove that, in ZFC, there exists a non-meager Lebesgue null subgrooup of R, while it isconsistent there there is no non-null meager subgroup of R.
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.
Assuming four strongly compact cardinals, it is consistent that all entries in Cichoń's diagram are pairwise different, more specifically that \[ \aleph_1 < \mathrm{add}(\mathrm{null}) < \mathrm{cov}(\mathrm{null}) < \mathfrak{b} < \mathrm{non}(\mathrm{meager}) < \mathrm{cov}(\mathrm{meager}) < \mathfrak{d} < \mathrm{non}(\mathrm{null}) < \mathrm{cof}(\mathrm{null}) < 2^{\aleph_0}.\]
In [arXiv:1605.02261] Rosłanowski and Shelah asked whether every locally compact non-discrete group has a null but non-meager subgroup, and conversely whether it is consistent with $ZFC$ that in every locally compact group there are no meager but non-null subgroups. The answer is affirmative for both questions [arXiv:1605.02261], however, in this paper we provide a much simpler proof for the special case of Abelian Polish groups.
In [arXiv:1605.02261] Rosłanowski and Shelah asked whether every locally compact non-discrete group has a null but non-meager subgroup, and conversely, whether it is consistent with $ZFC$ that in every locally compact group a meager subgroup is always null. They gave affirmative answers for both questions in the case of the Cantor group and the reals. In this paper we give affirmative answers for the general case.
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.
For a Polish group G let cov_G be the minimal number of translates of a fixed closed nowhere dense subset of G required to cover G. For many locally compact G this cardinal is known to be consistently larger than cov(meager) which is the smallest cardinality of a covering of the real line by meagre sets. It is shown that for several non-locally compact groups cov_G=cov(meager). For example the equality holds for the group of permutations of the integers, the additive group of a separable Banach space with an unconditional basis and the group of homeomorphisms of various compact spaces. Most re…
We describe a simple machinery which translates results on algebraic sums of sets of reals into the corresponding results on their cartesian product. Some consequences are: 1. The product of a meager/null-additive set and a strong measure zero/strongly meager set in the Cantor space has strong measure zero/is strongly meager, respectively. 2. Using Scheepers' notation for selection principles: Sfin(Omega,Omega^gp)\cap S1(O,O)=S1(Omega,Omega^gp), and Borel's Conjecture for S1(Omega,Omega) (or just S1(Omega,Omega^gp)) implies Borel's Conjecture. These results extend results of Scheepers and Mill…
We construct, under the assumption that union of less than continuum many meager subsets of R is meager in R, an additive connectivity function f:R-->R with Cantor intermediate value property which is not almost continuous. This gives a partial answer to a question of D. Banaszewski. We also show that every extendable function g:R-->R with a dense graph satisfies the following stronger version of the SCIVP property: for every a<b and every perfect set K between g(a) and g(b) there is a perfect subset C of (a,b) such that g[C] subset K and g|C is continuous strictly increasing. This property is…
The feeling that those two forcing notions-Cohen and Random-(equivalently the corresponding Boolean algebras Borel(R)/(meager sets), Borel(R)/(null sets)) are special, was probably old and widespread. A reasonable interpretation is to show them unique, or ``minimal'' or at least characteristic in a family of ``nice forcing'' like Borel. We shall interpret ``nice'' as Souslin as suggested by Judah Shelah [JdSh 292]. We divide the family of Souslin forcing to two, and expect that: among the first part, i.e. those adding some non-dominated real, Cohen is minimal (=is below every one), while among…
Rigid many-one degrees contain infinite antichains of $1$-degrees
本文证明了m-刚性的集合其许多一度包含无限个1度的反链，并展示了该结论在随机实数中的应用。
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