Related papers: Cardinal invariants associated with Hausdorff meas…

We consider measures which are invariant under a measurable iterated function system with positive, place-dependent probabilities in a separable metric space. We provide an upper bound of the Hausdorff dimension of such a measure if it is…

We study the cardinal invariants of measure and category after adding one random real. In particular, we show that the number of measure zero subsets of the plane which are necessary to cover graphs of all continuous functions maybe large…

Generalising a construction of Falconer, we consider classes of $G_\delta$-subsets of $\mathbb{R}^d$ with the property that sets belonging to the class have large Hausdorff dimension and the class is closed under countable intersections. We…

The cardinal invariants $ \mathfrak h, \mathfrak b, \mathfrak s$ of $\mathcal P (\omega)$ are known to satisfy that $\omega_1 \leq \mathfrak h \leq\min\{\mathfrak b, \mathfrak s\}$. We prove that all inequalities can be strict. We also…

Let $\mathcal{SN}$ be the strong measure zero $\sigma$-ideal. We prove a result providing bounds for $\mathrm{cof}(\mathcal{SN})$ which implies Yorioka's characterization of the cofinality of the strong measure zero. In addition, we use…

We show that the escaping sets and the Julia sets of bounded type transcendental entire functions of order $\rho$ become 'smaller' as $\rho\to\infty$. More precisely, their Hausdorff measures are infinite with respect to the gauge function…

Let us assume that we are given two metric spaces, where the Hausdorff dimension of the first space is strictly smaller than the one of the second space. Suppose further that the first space has sigma-finite measure with respect to the…

For a strongly inacessible cardinal $\kappa$, we investigate the relationships between the following ideals: - the ideal of meager sets in the ${<}\kappa$-box product topology - the ideal of "null" sets in the sense of [Sh:1004]…

We deal with some of problems posed by Monk and related to cardinal invariant of ultraproducts of Boolean algebras. We also introduce and investigate some new cardinal invariants.

We study probabilistic iterated function systems (IFS), consisting of a finite or infinite number of average-contracting bi-Lipschitz maps on R^d. If our strong open set condition is also satisfied, we show that both upper and lower bounds…

Motivated by the work of D. Y. Kleinbock, E. Lindenstrauss, G. A. Margulis, and B. Weiss, we explore the Diophantine properties of probability measures invariant under the Gauss map. Specifically, we prove that every such measure which has…

We show that the Continuum Hypothesis implies that for every $0<d_1\leq d_2<n$ the measure spaces $(\RR^n,\iM_{\iH^{d_1}},\iH^{d_1})$ and $(\RR^n,\iM_{\iH^{d_2}},\iH^{d_2})$ are isomorphic, where $\iH^d$ is $d$-dimensional Hausdorff measure…

For a map $T \colon [0,1] \to [0,1]$ with an invariant measure $\mu$, we study, for a $\mu$-typical $x$, the set of points $y$ such that the inequality $|T^n x - y| < r_n$ is satisfied for infinitely many $n$. We give a formula for the…

We characterize the situation of having many normal measures on a measurable cardinal. We show the plausibility of having many normal measures on each compact cardinal.

We consider infinite Galton-Watson trees without leaves together with i.i.d.~random variables called marks on each of their vertices. We define a class of flow rules on marked Galton-Watson trees for which we are able, under some algebraic…

In this article we study the generalized Fourier dimension of the set of Liouville numbers $\mathbb{L}$. Being a set of zero Hausdorff dimension, the analysis has to be done at the level of functions with a slow decay at infinity acting as…

A recent article of Chernikov, Hrushovski, Kruckman, Krupinski, Moconja, Pillay and Ramsey finds the first examples of simple structures with formulas which do not fork over $\emptyset$ but are universally measure zero. In this article we…

We establish a formula yielding the Hausdorff measure for a class of non-self-similar Cantor sets in terms of the canonical covers of the Cantor set.

In this article, we study combinatorial properties of a certain ideal on $\omega$, called the \emph{Splitting ideal}. We calculate its cardinal invariants and its position in the Kat\v{e}tov order among other definable ideals. We also study…

Suppose $n\geq 2$ and $\mathcal{A}_{i}\subset \{0,1,\cdots ,(n-1)\}$ for $ i=1,\cdots ,l,$ let $K_{i}=\bigcup\nolimits_{a\in \mathcal{A}_{i}}n^{-1}(K_{i}+a)$ be self-similar sets contained in $[0,1].$ Given $ m_{1},\cdots ,m_{l}\in…