Models in which every nonmeager set is nonmeager in a nowhere dense Cantor set

These informal notes briefly discuss various aspects of Cantor sets.

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 (respectively, countably perfectly null) in $X$, if for every perfect Polish topology $\tau$ on $X$, giving the original Borel structure of $X$, $A$ is covered by an $F_\sigma$-set $F$ in $X$ with the original Polish topology such that $F$ is mea...

We introduce a notion of density point and prove results analogous to Lebesgue's density theorem for various well-known ideals on Cantor space and Baire space. In fact, we isolate a class of ideals for which our results hold. In contrast to these results, we show that there is no reasonably definable selector that chooses representatives for the equivalence relation on the Borel sets of having countable symmetric difference. In other words, there is no notion of density which...

We study the ideal of meager sets and related ideals.

This paper proves the existence of nonmeasurable dense sets with additional properties using combinatorial techniques.

We provide new techniques to construct sets of reals without perfect subsets and with the Hurewicz or Menger covering properties. In particular, we show that if the Continuum Hypothesis holds, then there are such sets which can be mapped continuously onto the Cantor space. These results allow to separate the properties of Menger and $\mathsf{S}_1(\Gamma,\mathrm{O})$ in the realm of sets of reals without perfect subsets and solve a problem of Nowik and Tsaban concerning perfec...

In this paper we define a new class of metric spaces, called multi-model Cantor sets. We compute the Hausdorff dimension and show that the Hausdorff measure of a multi-model Cantor set is finite and non-zero. We then show that a bilipschitz map from one multi-model Cantor set to another has constant Radon-Nikodym derivative on some clopen. We use this to obtain an invariant up to bilipschitz homeomorphism.

The ternary Cantor set $C$, constructed by George Cantor in 1883, is probably the best-known example of a perfect nowhere-dense set in the real line, but as we will see later, it is not the only one. The present article will delve into the richness and the peculiarities of $C$ through the exploration of several variants and generalizations and will provide an example of a non-centered asymmetric Cantor-like set.

We show that the Dual Borel Conjecture implies that ${\mathfrak d}> \aleph_1$ and find some topological characterizations of perfectly meager and universally meager sets.

In relation to the Erd\H os similarity problem (show that for any infinite set $A$ of real numbers there exists a set of positive Lebesgue measure which contains no affine copy of $A$) we give some new examples of infinite sets which are not universal in measure, i.e. they satisfy the above conjecture. These are symmetric Cantor sets $C$ which can be quite thin: the length of the $n$-th generation intervals defining the Cantor set is decreasing almost doubly exponentially. Fu...