Cohen real or random real: effect on strong measure zero sets and strongly meager sets

A Haar meager set that is not strongly Haar meager

Following Darji, we say that a Borel subset $B$ of an abelian Polish group $G$ is Haar meager if there is a compact metric space $K$ and a continuous function $f : K \to G$ such that the preimage of the translate, $f^{-1}(B+g)$ is meager in $K$ for every $g \in G$. The set $B$ is called strongly Haar meager if there is a compact set $C \subseteq G$ such that $(B+g) \cap C$ is meager in $C$ for every $g \in G$. The main open problem in this area is Darji's question asking whether these two notions are the same. Even though there have been several partial results suggesting a positive answer, in…

Shattered Iterations

We develop iterated forcing constructions dual to finite support iterations in the sense that they add random reals instead of Cohen reals in limit steps. In view of useful applications we focus in particular on two-dimensional "random" iterations, which we call shattered iterations. As basic tools for such iterations we investigate several concepts that are interesting in their own right. Namely, we discuss correct diagrams, we introduce the amalgamated limit of cBa's, a construction generalizing both the direct limit and the two-step amalgamation of cBa's, we present a detailed account of cB…

Small sets of reals through the prism of fractal dimensions

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.

关于猜测序列的渐进行为

On the Asymptotic Behavior of Guessing Sequences

本文研究了猜测序列的渐进行为，证明了渐近增长与概率一集猜测的关系，并通过随机游走和Cohen forcing构造了不同性质的猜测集。

How special are Cohen and random forcings i.e. Boolean algebras of the family of subsets of reals modulo meagre or null

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…

Products of special sets of real numbers

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…

Borel images of sets of reals

The main goal of this paper is to generalize several results concerning cardinal invariants to the statements about the associated families of sets. We also discuss the relationship between the additive properties of sets and their Borel images. Finally, we present estimates for the size of the smallest set which is not strongly meager.

自然数上模理想的Lebesgue测度零集

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。

拓扑 Erdős 相似性猜想与强零测集

Topological Erdős similarity conjecture and strong measure zero sets

本文证明实数集上拓扑普适集等价于强零测集，并推广至局部紧 Polish 群，从而确立不可数拓扑普适集存在性对 ZFC 的独立性，还提出全测度普适 Erdős 相似性猜想。