Related papers: Strongly meager sets do not form an ideal
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).
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…
We show that an ideal $\mathcal{I}$ on the positive integers is meager if and only if there exists a bounded nonconvergent real sequence $x$ such that the set of subsequences [resp. permutations] of $x$ which preserve the set of…
A set of reals A is called perfectly meager if A \cap P is meager in P, for every perfect set P. Marczewski asked if the product of perfectly meager sets is perfectly meager. In the paper it is shown that it is consistent that the answer to…
If ZFC is consistent, then each of the following are consistent with ZFC + 2^{{aleph_0}}= aleph_2 : 1.) X subseteq R is of strong measure zero iff |X| <= aleph_1 + there is a generalized Sierpinski set. 2.) The union of aleph_1 many strong…
We construct several models where there are no strongly meager sets of size continuum. In particular, there are no such sets in the Laver's model.
We show that not every family of generalized microscopic sets forms an ideal. Moreover, we prove that some of these families have some weaker additivity properties and some of them do not have even that.
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…
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…
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…
We study a strengthening of the notion of a perfectly meager set. We say that that a subset $A$ of a perfect Polish space $X$ is countably perfectly meager in $X$, if for every sequence of perfect subsets $\{P_n: n \in {\mathbb N}\}$ of…
By the Galvin-Mycielski-Solovay theorem, a subset $X$ of the line has Borel's strong measure zero if and only if $M+X\neq\mathbb{R}$ for each meager set $M$. A set $X\subseteq\mathbb{R}$ is meager-additive if $M+X$ is meager for each meager…
We present two constructions of gamma-sets which are large in sense of category.
A graph is strongly perfect if every induced subgraph H has a stable set that meets every nonempty maximal clique of H. The characterization of strongly perfect graphs by a set of forbidden induced subgraphs is not known. Here we provide…
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…
Let $S$ be the set of subsequences $(x_{n_k})$ of a given real sequence $(x_n)$ which preserve the set of statistical cluster points. It has been recently shown that $S$ is a set of full (Lebesgue) measure. Here, on the other hand, we prove…
A set $X \subseteq 2^\omega$ with positive measure contains a perfect subset. We study such perfect subsets from the viewpoint of computability and prove that these sets can have weak computational strength. Then we connect the existence of…
One says that a pair of sets $(S,Q)$ in $\mathbb{R}$ is 'annihilating' if no function can be concentrated on $S$ while having its Fourier transform concentrated on $Q$. One uses to distinguish between weak and strong annihilation types. It…
We study a strengthening of the notion of a universally meager set and its dual counterpart that strengthens the notion of a universally null set. We say that a subset $A$ of a perfect Polish space $X$ is countably perfectly meager…
We will show that there is no ZFC example of a set distinguishing between universally null and perfectly meager sets.